Given a quasiconformal mapping f:Rn→ Rn with n ≥ 2,we show that the(un-)boundedness of the composition operator Cf on the spaces Qα(Rn)depends on the index α and the degeneracy set of the Jacobian Jf.We establish sharp results in terms of the index α and the local/global self-similar Minkowski dimension of the degeneracy set of Jf.This gives a solution to [3,Problem 8.4] and also reveals a new phenomenon,which is completely different from the known results for Sobolev,BMO,Triebel-Lizorkin and Besov spaces.Tukia-V(a)is(a)l(a)s quasiconformal extension f:Rn → Rn of an arbitrary quasisymmetric mapping g:Rn-p → Rn-p is shown to preserve Qα(Rn)for all 0 ＜ α ＜ 1 when p ∈ [2,n)and 0 ＜ α ＜ 1/2 when p = 1.Additionally,Qα(Rn)is shown to be invariant under inversions for all 0 ＜ α ＜ 1.