[superb-owl]

You're looking for the Bell Experiment: https://en.wikipedia.org/wiki/Bell_test

It proves there's no "local hidden variable"--the state is indeterminate until they are measured. It's proven through some pretty simple probability theory, and is (relatively) easy to follow. There's a great video of Leonard Susskind explaining it somewhere

[stouset]

Imagine I send you a box. The box has three buttons: one on the top, one on the front, and one on the side. You can press only one of these buttons. When pressed it will either light up green or red. Pressing other buttons afterward causes no effect, the box is disabled.

I send your friend a copy of the same box. I tell you that the result of pressing each button is random but no matter what, if you both press the same button you will see the same result. If you press two different buttons, the results will be uncorrelated.

You ask me to send you and your friend a bunch of these paired boxes and start testing. You then both press random buttons on each box and record your results.

Comparing notes afterward you can see that every time you happened to press the same button you received the same result. You confirm that if you pressed two different buttons, you get uncorrelated results. No problem, you think. I have obviously preprogrammed each box to be one of GGG, GGR, GRG, GRR, RGG, RGR, RRG, or RRR.

But then you notice something strange. If this theory was true, for 2/8 boxes you would have an RRR/GGG box and would see the same answer no matter what. The remaining 6/8 boxes you should get the same answer 2/3 of the time. This means that your answers should agree 3/4 of the time. Even if you surmise that I never send you a box set to the same three values, your results should agree 2/3 of the time.

You crunch the numbers and find that your results agree precisely 50% of the time.