Multipartite nonlocal quantum correlations resistant to imperfections

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of superluminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in Einstein-Podolsky-Rosen (EPR) experiments in the presence of noise. We apply our method to an example involving $n$ parties sharing a Greenberger-Horne-Zeilinger-type state on which they carry out local measurements. For this example, we show that for local hidden-variable theories to reproduce the quantum correlations, the amount of superluminal classical communication $c$ and the detector efficiency $\eta$ are constrained by $\eta 2^{-c∕n}⩽O\left(n^{-1∕6}\right)$. This result holds even if the classical models are allowed to make an error with constant probability.