  We show that $A_s(n)$, the coordinate algebra of Wang's quantum permutation
group, is Calabi-Yau of dimension $3$ when $n\geq 4$, and compute its
Hochschild cohomology with trivial coefficients. We also show that, for a
larger class of quantum permutation algebras, including those representing
quantum symmetry groups of finite graphs, the second Hochschild cohomology
group with trivial coefficients vanishes, and hence these algebras have the AC
property considered in quantum probability: all cocycles can be completed to a
Sch\"urmann triple.
