Measurement sharpness cuts nonlocality and contextuality in every physical theory

Gathering data through measurements is at the basis of every experimental science. Ideally, measurements should be repeatable and, when extracting only coarse-grained data, they should allow the experimenter to retrieve the finer details at a later time. However, in practice most measurements appear to be noisy. Here we postulate that, despite the imperfections observed in real life experiments, there exists a fundamental level where all measurements are ideal. Combined with the requirement that ideal measurements remain so when coarse-grained or applied in parallel on spacelike separated systems, our postulate places a powerful constraint on the amount of nonlocality and contextuality that can be found in an arbitrary physical theory, bringing down the violation of Bell and Kocher-Specker inequalities near to its quantum value. In addition, it provides a new compelling motivation for the principles of Local Orthogonality and Consistent Exclusivity, recently proposed for the characterization of the quantum set of probability distributions.