I think entanglement gets a bit less strange if you just view it as correlation in a theory in which the sum of any two states of a physical system again is a state of that system (that is, quantum mechanics). What I mean by correlation is something like the following: let’s say I have a red marble, and a green marble, and two boxes to put them in, box A and box B.
Now, the setup of the system entails the following: whenever the red marble is in box A, the green marble is in box B; or conversely, whenever the green marble is in box A, the red marble is in box B. The marble colors are thus (anti-)correlated: once you know the color of one marble, you know the color of the other. There are thus two possible states for the system of the boxes and marbles to be in, which we can represent in the following somewhat curious way: |green[sub]A[/sub], red[sub]B[/sub]> or |red[sub]A[/sub], green[sub]B[/sub]>. Here, the indices simply denote the box, and the labels the color of the marble (the notation is a bit redundant: often, the index is omitted, and just the placement is used to indicate the ‘box’: first slot is ‘box A’, second slot is ‘box B’; but I’ll keep the indices for clarity).
Thus far the story in the classical world. But in quantum mechanics, as I said, things are a bit different: for any set of valid states, their sum is again a valid state. That is, if we just consider a single box for the moment, say box A, additionally to the states |green[sub]A[/sub]> and |red[sub]A[/sub]>, states of the form a|green[sub]A[/sub]> + b|red[sub]A[/sub]>, with certain constants a and b, are again valid system states. In the context of an experiment, i.e. if we open the box, the numbers a and b determine the probabilities of finding the marble inside to be either color; it will be green with probability |a|[sup]2[/sup], and red with probability |b|[sup]2[/sup]. Thus, if a = b = 1/√2, we will find either color with 50% probability. Note that this is not the case in the classical world: me wearing socks is a valid state, as is me wearing no socks. But me wearing socks and not wearing socks doesn’t make sense.
Now, the same considerations apply to systems of more than one box: as |green[sub]A[/sub], red[sub]B[/sub]> and |red[sub]A[/sub], green[sub]B[/sub]> are valid system states, so too is
|S[sub]AB[/sub]> = 1/√2(|green[sub]A[/sub], red[sub]B[/sub]> + |red[sub]A[/sub], green[sub]B[/sub]>)
What does it now mean for the system to be in this weird kind of state? Well, operationally, it means the following: if we open box A, and find the marble inside to be green, which will happen with a probability of 50%, then we will know instantly that the marble in box B is red; analogously, if we find the box A marble to be red (again with a probability of 50%), we know that the box B marble must be green.
This is what we observe in experiments on entangled quantum systems. Thus, entanglement is ultimately the superposition of correlated states—the mystery, then, comes ultimately from the superposition principle. From this point of view, entanglement ‘across time’ is maybe not so strange—certainly, events at two points in time can be classically correlated, even though they don’t co-occur. You could consider a set of four boxes (A, B, C, D), each set of two—A,B and C,D—correlated as above. You look into box A, note the color of the marble inside, then cast it to the flames.
Then, you fix the content of box C (which is only created at this point in time) such that it contains a marble of the same color as the one in B. Because of the correlation between boxes C and D, now box D must contain a marble whose color is opposite to that of A; thus, we have boxes A and D exhibiting the same correlation as the original A and B did, i.e. upon looking into box D, you will always find the color red if in the now destroyed you had found green, and green if you had found red. That is, they behave as if they were in either the state |green[sub]A[/sub], red[sub]D[/sub]> or |red[sub]A[/sub], green[sub]D[/sub]>. Add again superposition to the picture, and you can make them behave as if they were in the state |S[sub]AD[/sub]> as above.