  Quantum networks play a major role in long-distance communication, quantum
cryptography, clock synchronization, and distributed quantum computing.
Generally, these protocols involve many independent sources sharing
entanglement among distant parties that, upon measuring their systems, generate
correlations across the network. The question of which correlations a given
quantum network can give rise to, remains almost uncharted. Here we show that
constraints on the observable covariances, previously derived for the classical
case, also hold for quantum networks. The network topology yields tests that
can be cast as semidefinite programs, thus allowing for the efficient
characterization of the correlations in a wide class of quantum networks, as
well as systematic derivations of device-independent and experimentally
testable witnesses. We obtain such semidefinite tests for fixed measurement
settings, as well as parties that independently choose among collections of
measurement settings. The applicability of the method is demonstrated for
various networks, and compared with previous approaches.
