  Which is the simplest logical structure for which there is quantum
nonlocality? We show that there are only three bipartite Bell inequalities with
quantum violation associated with the simplest graph of relationships of
exclusivity with a quantum-classical gap. These are the most elementary logical
Bell inequalities. We show that the quantum violation of some well-known Bell
inequalities is related to them. We test the three Bell inequalities with pairs
of polarization-entangled photons and report violations in good agreement with
the quantum predictions. Unlike other experiments testing noncontextuality
inequalities with pentagonal exclusivity, the ones reported here are free of
the compatibility loophole.
