Homological properties of quantum permutation algebras

We show that $A_s(n)$, the coordinate algebra of Wang’s quantum permutation group, is Calabi-Yau of dimension $3$ when $n\ge 4$, 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.