The easiest example is the small-angle limit of the classical Bell inequality, which is probably the case where Bell noticed the violation first, since he mentions it explicitly as a particularly intuitive limit in his paper.
The situation is this: I have a machine that sends out two particles to two far away detectors. I can measure a yes/no quantity in 1 of 3 directions A,B,C. I notice the following:
- A,B,C always give the same answer on the two particles, so that they are 100% correlated.
- If I measure A on one particle and B on the other, the results are 99% correlated, so that 1% of the answers are different.
- If I measure B on one particle and C on the other, the results are 99% correlated, so taht only 1% of the answers are different between B and C.
Thinking classically, there are some variables determining the outcome of the measurements. Whatever these variables, they are the same in the two particles regarding the outcome of A,B,C (so as to reproduce point 1), and they are 99% correlated between the outcome of A and B, and between the outcomes of B and C.
This means that 99% of the time, the variable for A is the same as the variable for B, and 99% of the time the variable for B is the same as the variable for C. The largest fraction of the time that C can be different from A is then 98%, assuming that all the mismatching situations are disjoint.
But in quantum mechanics, when you do this, the results are 96% correlated! This might look like a small violation, but it is double the maximum allowed mismatch, so it's proportionately a big violation.
The experimental way to set this up is to consider a spin state of two electrons where they are fully entangled in a spin-singlet, so that the measurement of spin in three nearby directions A,B,C are always opposite. In order to make the situation exactly parallel to what I said above, you have to flip the results on one of the particles to make them 100% correlated instead of anti-correlated.
Now measuring one particle collapses the other to a definite spin state, and the probability of getting a mismatched spin in direction $\theta$ goes as the |sin(\theta/2)|^2 which goes as $|\theta^2|$. if you adjust the $\theta$ to make the probability of A and B mismatching 1%, you have the same $\theta$ between B and C, and so the angle between A and C is doubled. When you double $\theta$, you get 4 times the mismatch, because the probability goes as the angle squared.
From this you see that the problem is fundamental to the squaring of amplitudes, and any fix would require a mismatch in probability which is a sharp-turn type thing like $|\theta|$, where doubling the angle doubles the mismatch. This is clearly incompatible with the basis rotational invariance in quantum mechanics.
This is the intuitively simplest violation, in the spirit of Bell. Mermin presented things much in this way, but the small angle limit is important, because it makes the mathematics obvious as day.