In the EPR experiment, it is said that there are predictions that are made if there are hidden variables determined by something locally in each particle that results in a correlation in entanglement. And that these predictions don’t agree with the predictions in QM.
Do these same predictions apply even if the spin changes multiple times during its travel before reaching a detector? Or does it only apply if this property is constant once it’s emitted from the source?
This is best answered by using the idea of causality in special relativity. At each event there is a region of spacetime, called the backwards light cone, which contains all previous events that could possibly send an influence to that event. The Bell inequality applies to events not happening at the same place and which do not permit communication from one to the other at speeds less than or equal to light speed. Such events are said to be spacelike-separated. Bell's mathematical formulation considers a set of correlations between observations at the two events. To observe such correlations in practice one has to imagine the scenario is repeated many times with the same setup. The argument then shows there is a limit on the correlations if they are brought about by communication of classical information from a common event in the past light cones. The classical information can be changing with time as long as the probability distribution describing the possible observations at any event A does not depend on variables (hidden or otherwise) which can be influenced by events spacelike-separated from A. In other words, within the restriction of
describing physical situations using probability distributions (not wavefunctions)
ordinary limits on communication (as implied by special relativity)
one infers a restriction on a set of correlations. Quantum theory predicts the restriction can be broken, and observations agree with quantum theory.
Thus, Bell's argument does cover dynamic hidden variables, not just static ones. It's a long time since I read the paper and I don't recall him directly pointing this out in the text, but I think what I have written here is contained implicitly in the beautiful argument he discovered.
In my more detailed answer to a related question I mention how the correlations have to match the outcome of the equation $\sin^2(A-B)$ where A is the alignment of Alice's analyser and B is the alignment of Bob's analyser, in order to match the predictions of Quantum Mechanics.
Notice that the quantum prediction $\sin^2(A-B)$ does not include a variable for the polarisation of the incoming photon quantum so its polarisation is not important other than its assumed to be random. Any theory that is local and uses hidden variables only has the information about the angle of the incoming photon and the angle of the local analyser, so it cannot reproduce the $\sin^2(A-B)$ correlation, as it does not know both A and B. It does not matter if the incoming photon has a constant polarisation all along its route or if it is constantly changing. Either way it does not help determine $\sin^2(A-B)$.
In my other answer, I set up a game and a device that only uses local information and manage to produce the correct correlations when the analysers have the same orientation, the opposite orientation and when they are 45 degrees apart, but it fails, (as any local theory will always fail) at the angles in between.
Clarification note: In the other answer I used the $\cos^2(A-B)$ correlation equation, because I was using the example that the entangled photons were orientated the same way while in this answer I used the $\sin^2(A-B)$ equation which is the most often used and assumes the entangled photons are orientated at 90 degrees to each other. The orientations of entangled photons to each, depends on the method used to prepare them.
Bell's theorem assumes
(1) that at the time of measurement, the measured quantities are described by stochastic variables: single numbers with probabilities,
(2) that the measurement devices or particles don't somehow have information about what measurement settings will be chosen in advance and
(3) that any communication between the systems being measured happens locally.
Given these constraints the correlations between the measured quantities satisfy the Bell inequalities. Quantum theory doesn't satisfy the Bell inequalities.
Just to add to Andrew’s correct answer: The EPR idea (pre-Bell) needs to be kept in mind at all times as well. They pointed out that entanglement should lead to perfect correlations when identical observations are performed.
So… if one of the particles is changing mid-flight, the other one must be changing the same way. And… it doesn’t matter when they are detected relative to each other either. So how could they really be changing mid flight and still be perfectly correlated?
The point is: Both the EPR perfect correlations AND the Bell Inequality violations must be simultaneously considered with any alternative hypothesis.
