Hierarchical certification of nonclassical network correlations

With the increased availability of quantum technological devices, it becomes more important to have tools to guarantee their correct nonclassical behavior. This is especially important for quantum networks, which constitute the platforms where multipartite cryptographic protocols will be implemented, and where guarantees of nonclassicality translate into security proofs. We derive linear and nonlinear Bell-like inequalities for networks, whose violation certifies the absence of a minimum number of classical sources in them. We do so, first, without assuming that nature is ultimately governed by quantum mechanics, providing a hierarchy interpolating between network nonlocality and full network nonlocality. Second we insert this assumption, which leads to results more amenable to certification in experiments.