This picture was described in Scott Aaronson's essay The Ghost in the Quantum Turing Machine in 2013, and claims that human free will is related to our choices being caused by (and/or causing) quantum bits from the initial state of the universe that first have macroscopic effects inside our brains, where all other observers must have purely Knightian uncertainty over such bits[*]. For it to be plausible, a few facts have to be true about human brain biology and cosmic background radiation:

  1. "Quantum uncertainty—for example, in the opening and closing of sodium-ion channels—can not only get chaotically amplified by brain activity, but can do so “surgically” and on “reasonable” timescales."
  2. All photons that impinge on human brains have quantum states that could not "be altered, maintaining a spacetime history consistent with the laws of physics, without also altering classical degrees of freedom in the photons’ causal past".

Have these questions been studied in the intervening years, and what have the results been? Note that the plausibility of the picture has been discussed before on LW, and I'm not interested in further discussing whether a priori it seems at all promising to link free will and Knightian uncertainty.

[*] This is a poor summary, I recommend reading the paper if you have time.

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This was a very speculative if exciting essay, and I don't believe that there has been any serious research done in this area, in part because it is unclear where one would start without having a better understanding of the measurement problem. Certainly online search comes up empty. I think the main value of this work is that a computer scientist and part-physicist (though Scott Aaronson would probably deny that he is the latter) can make a non-trivial contribution to the age-old philosophical questions of free will and consciousness.

I can't even imagine how one could design an empirical study of whether human free will is caused by quantum effects.

Creating a control group of humans in a parallel universe that runs on classical physics, is already a great technical challenge, but assuming we have done so... how would we test whether they have free will?

(More generally, how does one empirically check whether something that is undefined depends on something that is omnipresent?)

Even if we ignore the "quantum" part of the thing, how does one empirically test whether something has a free will or not? Do actual humans pass this test? All of them, or only the neurotypical ones?

I think you might have misunderstood the question: I'm primarily asking about work done on points 1 and 2 listed in the question test, which don't mention 'free will'.

The earliest correct answer I know of to the question of "how do we have free will?" comes from St. Augustine, except instead of free will vs. determinism it was free will vs. divine omniscience. God knowing the future, Augustine says, doesn't invalidate our free will, because the cause of the choice still lies within our power, and that's what matters.

So yeah, sorry, I guess you weren't interested in talking about whether this makes any sense in relation to free will, but it does seem relevant when something is about 1500 years out of date.

Though for human amplification of quantum noise, check out the work on perception of single photons.

I guess you weren't interested in talking about whether this makes any sense in relation to free will

Indeed, most of your comment is off-topic in a way that I asked comments not to be. If you want to discuss that point, please write your own post or shortform comment, or write a comment in one of the linked posts.