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The set of quantum-realisable assemblages, or abbreviated to quantum assemblages, are those that arise from multiple parties 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 always solvable. In this paper we introduce a hierarchy of tests where each level either determines non-membership of the set of quantum assemblages or is inconclusive. The higher the level of the hierarchy the better one can determine non-membership, and this hierarchy converges to a particular set of assemblages. Furthermore, this set to which it converges contains the quantum assemblages. Each test in the hierarchy is formulated as a semidefinite program. This hierarchy allows one to upper bound the quantum violation of a steering inequality and the quantum advantage provided by quantum EPR assemblages in a communication or information-processing task.<\/jats:p>","DOI":"10.22331\/q-2025-01-14-1591","type":"journal-article","created":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T15:50:40Z","timestamp":1736869840000},"page":"1591","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios"],"prefix":"10.22331","volume":"9","author":[{"given":"Matty J.","family":"Hoban","sequence":"first","affiliation":[{"name":"Quantum Group, Department of Computer Science, University of Oxford, United Kingdom"}]},{"given":"Tom","family":"Drescher","sequence":"additional","affiliation":[{"name":"University of Innsbruck, Department of Mathematics, A-6020 Innsbruck, Austria"}]},{"given":"Ana Bel\u00e9n","family":"Sainz","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, 80-309 Gda\u0144sk, Poland"},{"name":"Basic Research Community for Physics e.V., Germany"}]}],"member":"9598","published-online":{"date-parts":[[2025,1,14]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Albert Einstein, Boris Podolsky, and Nathan Rosen. ``Can quantum-mechanical description of physical reality be considered complete?''. 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