Bell's theorem: the simplest versionEdit
Quantum mechanics permits us to create the following scenario.
 Pairs of particles are launched in opposite directions.
 Each particle is subjected to one of three possible measurements (1, 2, or 3).
 Each time the two measurements are chosen at random.
 Each measurement has two possible results, indicated by a red or green light.
Here is what we find:
 If both particles are subjected to the same measurement, identical results are never obtained.
 The two sequences of recorded outcomes are completely random. In particular, half of the time both lights are the same color.
If this doesn't bother you, then please explain how it is that the colors differ whenever identical measurements are performed!
The obvious explanation would be that each particle arrives with an "instruction set" — some property that predetermines the outcome of every possible measurement. Let's see what this entails.
Each particle arrives with one of the following 2^{3} = 8 instruction sets:

 RRR, RRG, RGR, GRR, RGG, GRG, GGR, or GGG.
(If a particle arrives with, say, RGG, then the apparatus flashes red if it is set to 1 and green if it is set to 2 or 3.) In order to explain why the outcomes differ whenever both particles are subjected to the same measurement, we have to assume that particles launched together arrive with opposite instruction sets. If one carries the instruction (or arrives with the property denoted by) RRG, then the other carries the instruction GGR.
Suppose that the instruction sets are RRG and GGR. In this case we observe different colors with the following five of the 3^{2} = 9 possible combinations of apparatus settings:

 1—1, 2—2, 3—3, 1—2, and 2—1,
and we observe equal colors with the following four:

 1—3, 2—3, 3—1, and 3—2.
Because the settings are chosen at random, this particular pair of instruction sets thus results in different colors 5/9 of the time. The same is true for the other pairs of instruction sets except the pair RRR, GGG. If the two particles carry these respective instruction sets, we see different colors every time. It follows that we see different colors at least 5/9 of the time.
But different colors are observed half of the time! In reality the probability of observing different colors is 1/2. Conclusion: the statistical predictions of quantum mechanics cannot be explained with the help of instruction sets. In other words, these measurements do not reveal preexistent properties. They create the properties the possession of which they indicate.
Then how is it that the colors differ whenever identical measurements are made? How does one apparatus "know" which measurement is performed and which outcome is obtained by the other apparatus?
Whenever the joint probability p(A,B) of the respective outcomes A and B of two measurements does not equal the product p(A) p(B) of the individual probabilities, the outcomes — or their probabilities — are said to be correlated. With equal apparatus settings we have p(R,R) = p(G,G) = 0, and this obviously differs from the products p(R) p(R) and p(G) p(G), which equal What kind of mechanism is responsible for the correlations between the measurement outcomes?
 You understand this as much as anybody else!
The conclusion that we see different colors at least 5/9 of the time is Bell's theorem (or Bell's inequality) for this particular setup. The fact that the universe violates the logic of Bell's Theorem is evidence that particles do not carry instruction sets embedded within them and instead have instantaneous knowledge of other particles at a great distance. Here is a comment by a distinguished Princeton physicist as quoted by David Mermin^{[1]}

 Anybody who's not bothered by Bell's theorem has to have rocks in his head.
And here is why Einstein wasn't happy with quantum mechanics:

 I cannot seriously believe in it because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.^{[2]}
Sadly, Einstein (1879  1955) did not know Bell's theorem of 1964. We know now that

 there must be a mechanism whereby the setting of one measurement device can influence the reading of another instrument, however remote.^{[3]}
 Spooky actions at a distance are here to stay!
 ↑ N. David Mermin, "Is the Moon there when nobody looks? Reality and the quantum theory," Physics Today, April 1985. The version of Bell's theorem discussed in this section first appeared in this article.
 ↑ Albert Einstein, The BornEinstein Letters, with comments by Max Born (New York: Walker, 1971).
 ↑ John S. Bell, "On the Einstein Podolsky Rosen paradox," Physics 1, pp. 195200, 1964.