  We prove that K-polystable degenerations of Q-Fano varieties are unique.
Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is
separated. Together with [Jia17,BL18], the latter result yields a separated
Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of
fixed dimension and volume. The result also implies that the automorphism group
of a K-stable Q-Fano variety is finite.
