  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.
