Given a quasiconformal mapping $f:{\mathbb R}^n\to{\mathbb R}^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\mathbf C}f$ on the spaces $Q{\alpha}({\mathbb R}^n)$ depends on the index $\alpha$ and the degeneracy set of the Jacobian $J\_f$. We establish sharp results in terms of the index $\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J\_f$. This gives a solution to \[3, Problem 8.4] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel–Lizorkin and Besov spaces. Consequently, Tukia–Väisälä's quasiconformal extension $f:{\mathbb R}^n\to{\mathbb R}^n$ of an arbitrary quasisymmetric mapping $g:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p}$ is shown to preserve $Q\_{\alpha} ({\mathbb R}^n)$ for any $(\alpha,p)\in (0,1)\times\[2,n)\cup(0,1/2)\times{1}$. Moreover, $Q\_{\alpha}({\mathbb R}^n)$ is shown to be invariant under inversions for all $0<\alpha<1$.