  It is usually believed that coarse-graining of quantum correlations leads to
classical correlations in the macroscopic limit. Such a principle is known as
$macroscopic$ $locality$. In this letter we re-examine this principle and
consider a macroscopic Bell experiment in which parties measure collective
observables with a resolution of the order of $N^\alpha$, where $N$ is the
number of particles and $\alpha$ defines the level of coarse-graining. One
expects nonlocal correlations to fade away as $\alpha$ increases, so the
natural question to ask is: what is the value of $\alpha$ required to restore
locality, marking the quantum-to-classical transition? In a previous work [M.
Navascu\'{e}s and H. Wunderlich, Proc. R. Soc. A. $\textbf{466}$: 881 (2010)]
it was demonstrated that locality is restored at $\alpha=1/2$ in the case of
$independent$ $and$ $identically$ $distributed$ (IID) entangled pairs. Here we
show that this is no longer the case in the generic (non-IID) scenario, where
the Hilbert space structure of quantum theory, along with the superposition
principle and the Born rule, can survive in the macroscopic limit. This leads
straightforwardly to a Bell violation for macroscopic observables, and thus the
question of the quantum-to-classical transition remains open.
