It was believed until recently that the verification of quantum entanglement and quantum steering, between two parties, required trust in at least one of the parties and their devices, in contrast to the verification of Bell nonseparability. It has since been shown that this is not the case: the need for trust, in verifying two parties share a given quantum correlation resource, can be replaced by quantum refereeing, in which the referee sends quantum signals rather than classical signals to untrusted parties. The existence of such quantum-refereed games is discussed, with particular emphasis on how they make it impossible for the parties to cheat. The example of a particular quantum-refereed steering game is used to show explicitly how measurement-device independence is achieved via ‘quantum programming’ of untrusted measurement devices; how cheating is prevented by the steered party being unable to distinguish sufficiently well between two sets of nonorthogonal signal states; and that cheating remains impossible when one-way communication is allowed from the steered party to the steering party. This game has been recently implemented experimentally, and is of particular interest both in accounting for any imperfections in the referee’s preparation of signal states, and in suggesting the future possibility of secure two-sided quantum key distribution with Bell-local states.