Quantum measurement of spins and magnets, and the classical limit of PR-boxes

When Alice measures all her spin-$\text{half}$ of a large ensemble of $N$ singlets, all along the same direction $\vec{a}$, she prepares at a distance an ensemble of spins for Bob which, because of statistical fluctuations, have a magnetic moment of the order $\sqrt{N}$. By making $N$ large enough, this magnetic moment can be made arbitrarily large. We show that, nevertheless, Bob can’t read out of this large magnetic moment Alice’s choice of measurement direction $\vec{a}$. We also consider stronger than quantum correlations and show that Tsirelson’s bound follows from the physical assumption that in the macroscopic limit all measurements are compatible and that this should not lead to signaling.