Not sure if this question has been addressed before so apologies if so.

If I’m understanding what I read correctly, there is (very probably) no such thing as hidden variables- a quantum does not contain any sort of property that determines what the outcome of a measurement on it will be. I’m fine with that. But what about correlation itself- the fact that if one quantum is measured as + the other will always be - ? In other words, where does correlation come from, where does it end, and where does the property of “oppositeness” between the two reside in the meantime? What does it mean, other than to predict the outcome of measurements, to say that two quanta are entangled while two random other quanta are not?

I would refer to them as “particles” rather than “quanta”…

Anyway, the state of the particle(s) is given by some function \psi. Suppose there are two particles; then the total state will satisfy something like \psi(x,y) = \psi_1(x)\psi_2(y). However, if they are “entangled” then you will not be able to separate variables like that, namely you will have something like eg \psi(x,y) = \sin(7\pi x)\sin(\pi y) + \sin(5 \pi x)\sin(5 \pi y), now if you measure x it tells you something new about y

This absolute correlation (or anticorrelation) only happens when the two experiments both measure the exact same variable (for instance, both measuring the y component of the spin of a particle). And for some other pairs of variables, you get exactly zero correlation (for instance, measuring components of spins at right angles to each other). Both of those could be explained by hidden variables, and neither is all that interesting.

Where things get really interesting is in between those two cases: For instance, measuring the component of spin along two different axes at an acute angle to each other. In that case, you’ll get a correlation, but it’ll be only a partial correlation: Both measurements will usually be opposite of each other, but not always. What’s interesting about this is that Bell was able to calculate just exactly how much correlation you could get, in such cases, using hidden variables, plus a small set of other assumptions that seem reasonable. But quantum mechanics actually finds a correlation that’s larger than Bell’s maximum possible correlation. And experiment agrees with quantum mechanics, not with Bell’s calculation.

That’s what tells us that quantum mechanics doesn’t just run using hidden variables, and in fact must violate at least one of those reasonable-seeming assumptions.

In some ways, entanglement just means that something can’t happen, or that certain possibilities are less likely. Suppose you have two experiments that each have the outcome 0 or 1. We’ll represent two 0 measurements as |00> (so-balled bra-ket notation). The complete set of measurements is then (ignoring the overall scaling):
|00> + |01> + |10> + |11>

No entanglement there; each possible outcome has equal probability. But the set of measurements could be this instead:
|00> + |11>

In that case, the results of the two experiments are entangled (say, because they measured the spin of entangled particles).

There are intermediate possibilities as well. In the general case, our results could be:
a|00> + b|01> + c|10> + d|11>

Where a, b, c, and d are independent constants, subject to the constraint that |a|^{2}+|b|^{2}+|c|^{2}+|d|^{2} = 1 (i.e., the probabilities sum to 1).

The best answer IMO would be that the entanglement is a property of the system under study. You can see this illustrated well in the previous answers as they talk about the quantum state, which is the quantum description of the system. Of course the idea of an isolated system with two paricles entangled with each other and nothing else is a useful fiction. That system is an inseperable part of a larger system and so on, until you get to the level of the Universe as a whole. That’s why the basic idea of many worlds theory is quite natural one in quantum mechanics.

There aren though hidden variable theories consistent with quantum mechanics, the most famous of these is Bohmian mechanics. In Bohmian mechanics the correlation doesn’t arise from the properties of particles, it arises from the quantum potential. The quantum potential is non-local and is defined on the configuration space of the system. Even in a hidden variables theory, entanglement remains a non-localized property of the system as a whole.