Device-independent secret sharing and a stronger form of Bell nonlocality

Bell nonlocality, the fact that local hidden-variable models cannot reproduce the correlations obtained by measurements on entangled states, is a cornerstone in our modern understanding of quantum theory. Apart from its fundamental implications, nonlocality is also at the core of device-independent quantum information processing, the successful implementation of which is achieved without precise knowledge of the physical apparatus. Here we show that a stronger form of Bell nonlocality, for which even some nonlocal hidden-variable models cannot reproduce the quantum predictions, allows us to circumvent possible attacks in the implementation of secret sharing, a paradigmatic communication protocol in which a secret split amid many possibly untrusted parts can be decoded only if they collaborate among themselves.