The quantum maxima for the basic graphs of exclusivity are not reachable in Bell scenarios

A necessary condition for the probabilities of a set of events to exhibit Bell non-locality or Kochen-Specker contextuality is that the graph of exclusivity of the events contains induced odd cycles with five or more vertices, called odd holes, or their complements, called odd antiholes. From this perspective, events whose graph of exclusivity are odd holes or antiholes are the building blocks of contextuality. For any odd hole or antihole, any assignment of probabilities allowed by quantum theory can be achieved in specific contextuality scenarios. However, here we prove that, for any odd hole, the probabilities that attain the quantum maxima cannot be achieved in Bell scenarios. We also prove it for the simplest odd antiholes. This leads us to the conjecture that the quantum maxima for any of the building blocks cannot be achieved in Bell scenarios. This result sheds light on why the problem of whether a probability assignment is quantum is decidable, while whether a probability assignment within a given Bell scenario is quantum is, in general, undecidable. This also helps to understand why identifying principles for quantum correlations is simpler when we start by identifying principles for quantum sets of probabilities defined with no reference to specific scenarios. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.