When you learned that apples are composed of atoms, did you also conclude that apples are just an illusion, so there is no point in eating them?

I guess the answer here is that apples are still real on a macroscopic level, and everything you learned about them remains true. On a microscopic level, there are just configurations of particles. These particles, in some configurations, create what we perceive as an apple.

To understand the composition of the apple, you have to divide it into smaller parts. Those smaller parts are not apples.

Now apply the same reasoning to time.

For all the usual purposes, time is real. If you want to understand "how the time was built", you have to find the underlying mechanism, and whatever that mechanism might be, it is not time. (Because, generally, the explanation for X is not X. Otherwise we would call it circular logic.)

For example, we could model time (ignoring quantum physics) as a set of "possible moments", which are connected by arrows according to the laws of physics. In this model, the "flow of time" would kinda mean moving your finger along the arrows, starting from an arbitrary point, and watching how the situation evolves... except that from your perspective, nothing really evolves, all those possible moments are frozen, it's just the movement of your finger that points at different moments.

The obvious question in this model is: given that the laws of physics are reversible on small scale, why do these moments "contain information" about their past, but not about their future? We could just as easily move the finger in the opposite direction.

One problem with this model is that it ignores quantum physics. If you add it to the model, you have multiple arrows going from one moment into the future. Moving your finger along the arrows means you need to choose randomly (the Copenhagen interpretation) or you need many fingers (the Many Worlds interpretation). This seems to give some answer: the past is unambiguous, the future is not.

Only, this is still not a correct model of quantum physics. In quantum physics we don't really have one state evolving into a collection of states, but rather a collection of states (with a complex amplitude each) evolving into a different collection of states (with a complex amplitude each). So in the metaphor of the finger moving between the moments, each "moment" would actually refer to something quite complex. And it would get even more complicated if we tried to add relativity to this model (what is a "moment" if there is no absolute time frame?).

Also, this model still feels wrong to me. If we imagine the set of all "possible moments" (possible configurations of particles in the universe), and then select anthropically for those where humans exist... then our universe still seems much more regular than a random selection of these moments. (People like to talk about Boltzmann brains, but I haven't heard a good explanation why a Boltzmann brain remembering an orderly past should be more likely than a Boltzmann brain remembering a chaotic past; it should actually be the other way round, isn't it?) Which means I am still fundamentally wrong about something... not sure what exactly.

Anyway, my point is that instead of calling time an "illusion", it would be proper to talk about time being composed of something that is not-time. (I don't know what that something might be, and I have no idea whether anyone does, but the simple models I have heard about are all wrong.) The fact that time is composed of something doesn't make it less real, just like apple being composed of atoms isn't less of an apple.

The psychological arrow seems to be mostly about memory (we remember the past, not the future), which is about information (human memory is just another information storage medium), which I guess is related to quantum entanglement and possibly some other things (such as our universe containing lots of useful energy). But this is not specifically about humans; the iron also rusts only in one direction of time.

Suppose Alice and Bob are building a three-story house, with a big spiral staircase running from floor 1 to floor 3.

Alice: Isn't it weird how we perceive this staircase as spiraling even though in reality, it's just sitting there?

Bob: What do you mean "perceive?" This is a spiral staircase. It spirals.

Alice: Not the shape of the whole staircase, I mean it seems like the stair goes in a spiral.

Bob: "The" stair?

Alice: Yeah, the stair. Right now I'm standing here on stair 15, but if I moved up to stair 16, the stair would seem to have gone in a spiral. Weird, right?

Bob: I still don't get this "perceive" thing. Of course when you go from stair 15 to stair 16 the stairs look like a spiral. That's because the stairs are a spiral.

Alice: Not the stairs, the stair! The stair that I'm on when I go from 15 to 16. It seems like it goes in a spiral. But there's actually no spiraling, and no stair. In reality it's just a bunch of stairs arranged in a helical shape. Sigh. It's so weird.

Moral of the story: Reality isn't weird. We're weird.

If I understand correctly the psychological arrow of time tries to explain why we perceive time as passing by. It answers that in truth time does not pass—we exist at a single point in the timeline, but have the illusion of time passing by because we remember the past and not the future

The arrow of time and the (at least apparent) passingness of time, are two different things. The natural numbers have an arrow, since they are well ordered, but that doesn't mean they have passingness.

We exist at a single point in the timeline,

So none of us is so much as nanosecond old?

Its often stated that physics is timeless...in some sense. But the senses could include lack of passingness, or lack of an arrow, or lack of a fundamental arrow, and so on.

Secondly isn’t memory itself a process that takes place over time? So how can the illusion occur if time isn’t passing in which it can occur?

If you are saying that passingness can't arise from stasis,I'm inclined to agree, although it's only intuition.

But we can explain how an arrow arises from fundamental symmetry, eg. the entropic explanation.

And the psychological arrow might be an uninteresting special case of the entropic arrow...or not.

we exist at a single point in the timeline, but have the illusion of time passing by because we remember the past and not the future.  is this approximately correct?

No.  It's not really coherent or testable enough to be correct or incorrect.  You can't know if "we" even "exist" in the sense you're describing.  I know I exist (per Descartes), but that's not evidence of any other part of your description.  It's possible that there is no continuity to my existence (see https://en.wikipedia.org/wiki/Boltzmann_brain for an example of how this could be).  

A simpler description (for my modeling; I don't know how to assert any objective simplicity measure) is that time is NOT JUST psychological, it's an actual part of the universe, and the future is different from the past in at least the fact that it has higher entropy.   

Related: I highly recommend https://smile.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 as an introduction to some of the concepts about time and cognition.  Especially the part about the tortoise enjoying records (vinyl music recordings) by slapping them against his stomach to experience the pattern of grooves all at once.  

I think you might be confusing two things: the arrow of time, and the hard problem of consciousness.

The arrow of time refers to the fact that there as a difference between the past and the future.  This is straightforward, and has to do with the fact that entropy increases over time (2nd law of thermodynamics).  This also explains lots of things like why your brain has an easier time remembering the past than predicting the future.  In theory if we lived in a universe where entropy was at a maximum (and all laws of physics were reversible) there would be no obvious difference between the past and the future.

The hard problem of consciousness asks: why do humans perceive time as a series of moments (or more fundamentally, why do humans perceive anything at all)?  Because perception is inherently subjective, this question is beyond the scope of scientific inquiry.  This accounts for the hardness of the hard problem of consciousness.  Of course the fact that perception is subjective and hence fundamentally beyond the reach of science doesn't stop philosophers from speculating about it.  

Around here at Less Wrong, the theories you are most likely to come across is "time is an illusion" or "perception is what an algorithm feels like from the inside".  Of course both of these theories might be unsatisfying to you, since like all human beings you undoubtedly experience time as a series of moments, not as a timeless whole.  

Unfortunately, the competing theories (e.g. "time is created when conscious beings cause quantum waveform collapse") are all pretty bad, since they depend on the existence on some non-physical "consciousness substance".  These theories are collectively called dualism and were espoused by such greats as Rene Descartes but are now generally out of vogue.  The main problem with these theories is that the "consciousness substance" would itself be subject to metaphysical laws, and then we would again be stuck asking "but what about those laws causes consciousness to arise?" leaving us in some sort of endless regress.  Talking  about non-physical substances that can never be measured is also considered a big no-no under Occam's Razor.

There is a way out of this particular trap, often taken by Buddhist and Existentialist philosophers.  Namely, any time you ask them to explain consciousness, they shake their head and grumble "Existence is existence! It cannot be explained! It can only be experienced!"  While this neatly avoids the argument (by refusing to engage in it), it can certainly be frustrating if you want to understand what consciousness is.

I think it might be useful to think about what it would mean for the sentence “time is an illusion” to be true, or to be false.

There’s a certain contrarian perspective I find it useful to be able to take, by applying that perspective to this question I generated a few scripts:

“Of course a timeline is defined as a static object – if you take time out of the world-in-which-your-model-is-embedded and put it directly into your model you’ll find that there’s no time left over for your model to change in. There’s only one dimension of time, so you can’t have a model of time that changes, because if you did then I would have to ask exactly what your model changes over, if not time; once you take time and put it in the model, you can’t have time that operates outside the model!”

“A couple of your other confusions (if time doesn’t pass, how can you do anything? isn’t everything useless?) seem to be related to the free will/determinism conflict (if everything we do is already determined, what use is it making any decisions?). I occasionally find people that think that determinism constrains and prevents free will, because if your decision is just the result of initial conditions of the things you’re making the decision over, then how do you have the freedom to choose either option?, to which I reply, if you weren’t able to make the decision just based on everything you knew, what else would you posit is affecting it? What third-party source of noise has affected the process by which you are making decisions, so that all options are somehow possible?”

“The feeling-of-time-passing and the feeling-of-yourself-thinking are one and the same – you couldn’t alter one and not affect the other; the feeling-of-time-passing is the result of you moving to new mental states over new thoughts and new memories, as a result of being part of a chain of cause-and-effect that includes yourself as an element and as a subject. The feeling of making decisions and resolving sensory input into data, the feeling of entropy ratcheting forwards, is the feeling of time passing.”

“What would it mean for there to be a now that isn’t ‘subjectively’ defined by the you at each moment? Would that posit a perfect, timeless crystal that also happens to have a little arrow that says ‘you are here’, that moves forward at the rate of one crystal-second per second?”

“The time-that-is-now is the current position of the you within that timeless crystal, and all of them are correct – you can’t have a timeless crystal that has a ‘current time’ on it for the same reason that you can’t have a locationless map beamed onto the moon that also has a ‘you are here’ arrow on it.”

“There is no arrow of time – the more we understand physics, the more timeless our understanding becomes, but there IS an arrow of entropy (defined by taking a given starting state and running physics in either direction), and asking ‘what would it mean for use to go in the opposite direction from entropy, remembering our future but not our past’ yields a contradiction; the direction we experience entropy in is the direction we experience time passing in, because that's what it means to be affected by entropy."

I can’t put full epistemic emphasis on any of these individually, but I hope you can see the general line that I’m working from.

PBS Space Time addresses the memory question in a really accessible way here. The key insight is (briefly) that one end of a particular object's history is highly correlated with the environment along which its duration extends (i.e. you can infer information about other parts of the environment by examining the object at that end if its duration). The other end is not so correlated. The degree to which this correlation exists at a particular point along the object's duration is its "memory" at that particular "moment in time".

Psychological arrow of time is taking a different stance towards the future as opposed to past. It might be nice to try to explain but the arrow itself is this assymetry.

A brain might try to recall the past but it won't try to recall the future. You can predict the future but one seldom predicts the past. Similarly with goal oriented behaviour one can try to become a good person, one less frequently tries to have been a good person.

One can think of it as contents of mind being a comic strip and the panels containing a snapshot of the "primary information". However all the panels exist at the same time. Imagine the flowing process to be to destroy the leftmost panel and generate a new right panel. The panel migth get vandalised during its lifetime and would give a inaccurate representation of the "pictured" state. However "timeline as it appears right now" can be impressed instantly. If multiple pictures were vandalised simultanouesly then it could infact be vdery hard to detect. But the objects in the pictures are likely to display temporal inertia, looking at object in one picture is likely to tell and restraint a lot about what might be gooing on with the corresponding object in an adjacent panel. And this comparsion doesn't depend on whether it is carried out instantly or over a long period of time. Comic strips are completely frozen yet have a timeline in them.

You may exist at only a single point in the timeline, but very-slightly-different versions of you exist at nearby points (both nearby temporally and nearby via recent quantum branching). Physics determines the relationships between these versions, and also causally ties you into everything else. If you are wearing a hat at this instant, the versions of you a few milliseconds before and after are probably also wearing hats. Your memory is what it is because of what your past versions experienced.

You can think of "doing something" as "being the sort of person whose future self has done the thing" if you like, and you are yourself the sort of person you are based on what your past self was like (again via Physics), but the short answer here is "It all adds up to normality."

I'd say that the normal temporal dimension we impose on reality is related to, but not the same, as the kind of time that underpins our consciousness.

As you say, memory is a process, not a static snapshot; the act of being sentient cannot be usefully be broken down into a sequence of mind-states based at instances on the timeline.

But perhaps there can be something more like a dynamic snapshot; atomic slices of consciousness that span over normal time, and represent a combined state/process from which 'this moment, this thought, this feeling' can be abstracted. 

There's lots of ways to twist the kaleidoscope and interpret the underling structure, and they're all (of course!) related to each other