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Correlations in Einstein-Podolsky-Rosen (EPR) scenarios, captured by\textit{assemblages} of unnormalised quantum states, have recently caught theattention of the community, both from a foundational and aninformation-theoretic perspective. The set of quantum-realisable assemblages,or abbreviated to quantum assemblages, are those that arise from multipleparties performing local measurements on a shared quantum system. In general,deciding whether or not a given assemblage is a quantum assemblage, i.e.membership of the set of quantum assemblages, is a hard problem, and not alwayssolvable. In this paper we introduce a hierarchy of tests where each leveleither determines non-membership of the set of quantum assemblages or isinconclusive. The higher the level of the hierarchy the better one candetermine non-membership, and this hierarchy converges to a particular set ofassemblages. Furthermore, this set to which it converges contains the quantumassemblages. Each test in the hierarchy is formulated as a semidefiniteprogram. This hierarchy allows one to upper bound the quantum violation of asteering inequality and the quantum advantage provided by quantum EPRassemblages in a communication or information-processing task.

A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios