  Various protocols exist by which a referee can be convinced that two
observers share an entangled resource. Such protocols typically specify the
types of communication allowed, and the degrees of trust required, between the
referee and each observer. Here it is shown that the need for any degree of
trust of the observers by the referee can be completely removed via the referee
using classical and quantum communication channels appropriately. In
particular, trust-free verification of Bell nonlocality, EPR-steering, and
entanglement, respectively, requires two classical channels, one classical and
one quantum channel, and two quantum channels. These channels correspond to
suitable inputs of quantum randomness by the referee, which prevent the
observers from mimicking entanglement using shared classical randomness. Our
results generalize recent work by Buscemi [Phys. Rev. Lett. 108, 200401
(2012)], and offer a perspective on the operational significance of that work.
They also offer the possibility of simpler experimental demonstrations of the
basic idea of quantum-refereed nonlocality tests.
