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In this article, we derive a novel family of noncontextuality inequalities along with their sum-of-squares decompositions in the simplest (odd-cycle) sequential-measurement scenario capable to demonstrate Kochen-Specker contextuality. The sum-of-squares decompositions allow us to obtain the maximal quantum violation of these inequalities and a set of algebraic relations necessarily satisfied by any state and measurements achieving it. With their help, we prove that our inequalities can be used for self-testing of three-dimensional quantum state and measurements. Remarkably, the presented self-testing results rely on a single assumption about the measurement device that is much weaker than the assumptions considered in Kochen-Specker contextuality.<\/jats:p>","DOI":"10.22331\/q-2020-08-03-302","type":"journal-article","created":{"date-parts":[[2020,8,3]],"date-time":"2020-08-03T14:31:44Z","timestamp":1596465104000},"page":"302","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":20,"title":["Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4525-0903","authenticated-orcid":false,"given":"Debashis","family":"Saha","sequence":"first","affiliation":[{"name":"Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\u00f3w 32\/46, 02-668 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4695-530X","authenticated-orcid":false,"given":"Rafael","family":"Santos","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\u00f3w 32\/46, 02-668 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1154-6132","authenticated-orcid":false,"given":"Remigiusz","family":"Augusiak","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\u00f3w 32\/46, 02-668 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2020,8,3]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"B. 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