Simple explanation of the quantum limits of genuine $n$-body nonlocality

Quantum $n$-body correlations cannot be explained with $(n-1)$-body nonlocality. However, this genuine $n$-body nonlocality cannot surpass certain bounds. Here we address the problem of identifying the principles responsible for these bounds. We show that, for any $n\ge 2$, the exclusivity principle, as derived from axioms about sharp measurements, and a technical assumption give the exact bounds predicted by quantum theory. This provides a unified explanation of the bounds of single-body contextuality and $n$-body nonlocality, and connects two programs towards understanding quantum theory.