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Violation of a noncontextuality inequality or the phenomenon referred to`quantum contextuality' is a fundamental feature of quantum theory. In thisarticle, we derive a novel family of noncontextuality inequalities along withtheir sum-of-squares decompositions in the simplest (odd-cycle)sequential-measurement scenario capable to demonstrate Kochen-Speckercontextuality. The sum-of-squares decompositions allow us to obtain the maximalquantum violation of these inequalities and a set of algebraic relationsnecessarily satisfied by any state and measurements achieving it. With theirhelp, we prove that our inequalities can be used for self-testing ofthree-dimensional quantum state and measurements. Remarkably, the presentedself-testing results rely on a single assumption about the measurement devicethat is much weaker than the assumptions considered in Kochen-Speckercontextuality.

Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices