I'm missing something basic in understanding Bell's theorem

Question

I've read the wikipedia article, but I'm lacking some very basic understanding of entanglement probably.

If I understand correctly, the experiment that is repeated is not time dependent, right?

e.g. my understanding is this:

1. Alice and Bob are far apart, really far so that they (or the "particles") can't send each other's states/observations in a way that will violate faster than light information traveling.

2. They have made in advance a very large set of pre-entangled particles

3. They randomly select a particle, write down the "particle ID" in excel, then at some random time, they measure it's spin, in a machine that has a random angle, they don't know who measured first or what was the angle of the other

4. They expect that if there was some internal hidden identical "state machine" in both particles, that deterministically decides before they separated, how they will respond once measured, for every possible angle, then it will be some V shaped chart (or other, as Gill describes in his papers)

5. However, they notice a more "bell" shape which is violating Bell's inequalities, hence proving there are either no locality, no realism, or no free will.

Am I understanding it correctly? E.g. once you measure a particle in one side, the other side can be non simultaneous (I assume they have synchronized clock and are in the same inertial frame). E.g. they are light years away, so they can measure particles not exactly at the same time, as long as light can't travel between Alice and Bob in the time between measuring of the same particle.

If this the case, although I don't fully understand the math of the theorem, I do appreciate it intuitively, e.g. I can't think how to design a computer that will do it, I need to design two devices that will either "lock" in advance their agreed state for all possible angles, or be in a time based PRNG sync at some point, but then "lock" when the other is locked, both sound improbable to me.

However if the theorem is also requiring them to be measured at the "exact" same time, then I don't understand how is that different than an RSA token / Yubikey, where both are time based, and both show pseudo random results when measured within the same 30 seconds, correlated as long as their clock is correct, and both have the same algorithm and seed.

tl;dr

Does Bell theorem experiments need that the random angle measurement be done "at the same time" (I know that special relatively means there is no such thing, but assuming they have some sort of magically synchronized clocks and are in the same inertial frame, I'm pretending I know what it means). If so, why does such a time based pseudo random number generator also can't explain hidden variables

Or does the experiments can happen at random times, and as long as the time between measurements in points A and B of the same particle is far enough so that information can't reach the other side between the two measurements. (this makes it much easier to understand, the particle in point A can't know when to do the internal "random" loop, vs when to "lock on the result, forever")

Sorry if it took so many words to ask such a simple question, but sometimes, the obvious things are omitted from advanced articles as the writer assumes the reader understand the basics.

Answer 1

Does Bell theorem experiments need that the random angle measurement be done "at the same time"?

It doesn't matter at what times Alice and Bob perform their measurements, nor when they "decide" their measurement settings. If they are far enough, then you can use the results of the experiment to rule out local hidden variables, which is an added bonus.

In this regard, the situation is really no different than "measuring correlations" in a classical scenario. Think of two persons holding each one of a pair of coloured marbles, one white and one black, but not knowing which one has which. Whatever the distance between them, and regardless of when they look at the colour of their marble, their results will always be fully correlated. Nor would you expect otherwise. The quantum case works the same (except of course the correlations are "weirder").