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How to understand the set of correlations admissible in nature is oneoutstanding open problem in the core of the foundations of quantum theory. Herewe take a complementary viewpoint to the device-independent approach, andexplore the correlations that physical theories may feature when restricted bysome particular constraints on their measurements. We show that demanding thata theory exhibits {a composite} measurement imposes a hierarchy of constraintson the structure of its sets of states and effects, which translate to ahierarchy of constraints on the allowed correlations themselves. We moreoverfocus on the particular case where one demands the existence of a correlatedmeasurement that reads out the parity of local fiducial measurements. Byformulating a non-linear Optimisation Problem, and semidefinite relaxations ofit, we explore the consequences of the existence of such a parity readingmeasurement for violations of Bell inequalities. In particular, we show that incertain situations this assumption has surprisingly strong consequences,namely, that Tsirelson's bound can be recovered.

Correlations constrained by composite measurements