If we consider how it is possible to implement technological resurrection of people whose bodies have been lost (decomposed or cremated, so cryonics cannot be applied), two ways come to mind. The first one is some time manipulation. The second one is some kind of data analysis, computer modeling, retrodiction and reconstruction. Everything in between is also possible.
As for the first option: often, one can read on the Internet something like “according to laws of physics, time travel is impossible,” but it is wrong. Theoretically, time travel is possible, though we do not know how to overcome a few obstacles in the way of doing it yet.
Let us go into detail. So, according to Einstein’s general relativity theory, everything that has happened in the past still exists, but we cannot reach it through ordinary cinematic motion. We must either exceed the speed of light or change the curvature of the spacetime (the latter is curved, the curvature being determined by the mass/energy of matter). Wormholes (bridges for instant movement between distant regions) can also be used.
A number of time machine math models have been developed by theoretical physicists, the most promising being warp-drive based ones. Now, active work is conducted on 1) trying to figure out where to get “exotic matter” necessary for changing the curvature of the spacetime as well as how to reduce the amount of this matter; 2) trying to do without exotic matter and to reduce the amount of energy necessary for changing the curvature. So, we may now be on the verge of getting a warp-drive. This work is conducted, first of all, to be able to carry out interstellar flights, but such a device can also be used as a time machine. So, a means for technological resurrection can emerge as a by-product of the development of astronautics.
As far as the second option (computer modeling and information reconstruction with subsequent synthesizing a person to be resurrected), efforts are made in this area, too. There is a nascent field called “quantum archeology.” According to people who conceived it and work on it, “it seeks to reconstruct our past light cone, including the brain and body states of every person in history, with high fidelity through advances in physics and computer science.”
Roughly, it will look like that: a computer model of a relevant part of the universe will be implemented, the development of this system will be analyzed, and then “retrodiction” (prediction of the past) will be done. The data concerning a person to be resurrected will be used to bio-print the person using a molecular (at least) level precision 3D-printer.
This field included physics, IT and biotech. Now, work on the physical framework is being conducted. The IT part will be developed when sufficiently powerful computers (in all probability, quantum ones) will be at the disposal of the quantum archeology community. The biotech part will come last.
By the way, this community is present of Reddit, their subreddit being https://www.reddit.com/r/QuantumArchaeology/. Now, physicists are especially welcome. If you are a physicist and took interest in this, you can contact the subreddit moderator – he is a physicist and he will explain physics-specific things you are interested in. And if you want to contribute, you are warmly welcome. Telling other physicists of this project will also be beneficial for this cause.
Apart from working on the physical aspect, the community now is also spreading the word about quantum archeology and encourages those who support the cause to help do it. Biology and gerontology are fighting against ageing and death caused by it hoping to beat the enemy in the foreseeable future. Technological resurrection will make this victory absolute reclaiming even those who died shy of the defeat of the ageing related death as well as those who passed away because of diseases or accidents. Life will inevitably win.
As for comparing those two approaches, the former is better theoretically developed, more mature, whereas the latter is fit for mass resurrections and is probably cheaper.