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We argue that studying such objects leads to a better understanding of the relation between quantum nonlocality and non-locality beyond quantum mechanics. The main issue discussed in the paper is whether there exist &amp;apos;genuine&amp;apos; C-Q boxes or all C-Q boxes can be built from objects already known, namely C-C boxes acting on pre-shared entangled quantum particles. We show that large classes of C-Q boxes are non-genuine. In particular, we show that all bi-partite C-Q boxes with outputs that are pure states are non-genuine. We also present various strategies for addressing the general problem, i.e. for multi-partite C-Q boxes which output mixed states, whose answer is still open. Finally, we show that even some very simple non-genuine C-Q boxes require large amounts of C-C nonlocal correlations in order to simulate them.<\/jats:p>","DOI":"10.22331\/q-2024-08-22-1443","type":"journal-article","created":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T15:54:02Z","timestamp":1724342042000},"page":"1443","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Classical-to-quantum non-signalling boxes"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-5605-0643","authenticated-orcid":false,"given":"Carolina","family":"Moreira Ferrera","sequence":"first","affiliation":[{"name":"H. H. 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