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.
