> This is known as a Bell inequality. It captures the essential limitation imposed by any theory based on local hidden variables — theories that adhere to class

> This is known as a Bell inequality. It captures the essential limitation imposed by any theory based on local hidden variables — theories that adhere to classical notions of determinism (no random chance in the measurement apparatus), locality (no faster-than-light influences) and realism (pre-existing properties). Obligatory reminder that there is an extra assumption here: the assumption that the result of the coin flip is not correlated to the hidden state of the particle. If when receiving a particle in stage a_H your coin flip always leads to, say, HH, then you will break Bell's inequality even if all the other assumptions hold. Theories that have this property are called "superdeterministic".