Would Quantum entanglement theoretically allow prediction of the future? This article describes how a choice made by the recipient of an entangled photon can affect measurements taken on that photon's "partner" before the decision was made.
So let's say there are two recipients (A and B) receiving streams of paired photons. Recipient A chooses whether to combine certain photons based on some observation (like the movement of a particular stock ticker). 


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*Would this make it possible for recipient B to know how that stock ticker moved before it happened?

*If so, would it be possible for recipient B to have affected some of the photons they received based on their earlier observations, so that recipient A could know what would happen to the stock ticker before they even observed it?

 A: So: Can you use time entanglement to communicate across time?
Well, the simple (and correct0 answer is... no.
The rule always in entanglement is that you can't transfer information. Only after you examine both (or all) of the entangled answers can you determine that entanglement occurred. Individually, each entangled event looks perfectly random and gives no hint of events taking place at other locations.
That rule is usually applied to spatial entanglement, but the temporal version necessarily works the same way.
Why this is so is partly observation. But more deeply, the inability to transmit information using entanglement has deep ties both to causality and the nature of quantum mechanics. Quantum mechanics can be defined rather nicely as the physics of events that are truly and totally unknown anywhere in the universe -- "ahistorical" is a word I like to use sometimes to describe it. That is, quantum rules apply whenever you get situations where things are simple enough, small enough, cold enough, and/or protected enough so that no trace of information about how they work exists anywhere in the classical universe -- and really do mean "anywhere." Simply pretending you don't know the result is not enough.
When that situation happens, what occurs within that envelope of ignorance is that all possible histories are explored in away that shows up to us as waves. Even though universal ignorance of the final results from a particle view must be kept intact for such waves, the wave is very much visible and not at all abstract. These waves are, for example, the basis of those double-slit experiments that Richard Feynman described so beautifully in QED: The Strange Theory of Light and Matter. Poke a bit too hard at such waves, however, and they fall apart, producing a specific result that then becomes part of ordinary, information-rich "decided history."
Now if you think about all this, it's a beautiful symmetry of sorts. If you look closely, all forms of quantum behavior contain an element of entanglement. For example finding a photon in one location (say in a telescope on earth) out of an immense or even star-crossing wave representation of that same photon means that you must also "instantly" guarantee that no alien on a nearby star can see that same photon. That kind of trans-light consistency is a form of entanglement, of mass-energy rather than spin, but entanglement all the same.
So what happens is that quantum becomes the world of the ahistorical, that is, of the events (or pieces of events) that have not quite been resolved to produce a causal or historical result. Thus by definition they cannot violate the flow of history, even if they cause strange correlations across time itself. You can in the quantum world go back and shoot your own grandfather, but since your grandfather is allowed to be quantum only if there is absolutely no trace of what happened to him anywhere within the classical universe, the result is more like unwrapping a well-hidden package than like changing history. You can even exist to do it... provided that your grandfather had your mother before disappearing into quantum opacity. Otherwise you become one of the information traces of him, and he can no longer be quantum!
The other side of the symmetry is classical history, which is the abode of all those fully determined results for which evidence of their occurrence already exist somewhere within the universe. This classical fabric of the known makes it impossible to change something without violating some known past result -- a classic time paradox.
And in the case of your question, whether you phrase it in terms of time entanglement or spatial entanglement, the answer always works out the same: The entanglement can exist only if the earlier or most distant result is fully unknown. You can for example nominally change an event in the past in a way that makes it entangled with the present. But since you cannot know that past event, you either end up with it being "unwrapped" much later (no information transfer there; it just becomes another case of spatial entanglement), or the earlier event has already entered the historical record and predetermined how your test will proceed. Entanglement events unfolded over time always work out to be frustratingly symmetric, since you can never really say that one "caused" the other, only that they both look like either could have caused the other.
My standard recommendation for such questions is always the works of Richard Feynman, by the way. QED is a delightful read if you like weird thrown in your face, sans the math. To dive deeper, pretty much all of The Feynman Lectures on Physics: Volume III is a good resource for both the subtleties of how objects enter and leave the quantum world, and of the starting level mathematics (plus some advanced) of such things.
A: This is precisely why I'm not a fan of using the word "affect" when you're talking about entanglement. If you have two entangled photons, number 1 and number 2, then when you measure photon 1, it doesn't really affect photon 2, in the sense that the measurement of 1 does not induce a detectable change in photon 2. Your collaborator who is measuring photon 2 will get some result, but has no way of knowing whether that result came about because of random chance or because you already got the opposite result for photon 1.
The only thing that does happen because of the entanglement is that you the results of the measurements are correlated: when you and your collaborator get together and compare notes afterwards, you'll find certain relationships between the results of the measurements of photon 1 and the results of the measurements of photon 2. For example, one will always be polarized clockwise when the other is polarized counterclockwise, and vice versa. But if you look at the measurements of either photon by itself, without comparing results, everything looks completely random. For this reason, you can't use entanglement to transmit information.
In your situation, suppose A and B are receiving streams of entangled photons. Recipient B can measure the photons all he wants, but those measurements will reveal nothing about whether the photons are entangled with each other. And recipient A's measurements will reveal nothing about the results of recipient B's measurements.
