  We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano
cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e.,
Sasaki-Einstein manifolds with singularities.
  As a byproduct, it gives alternative algebraic construction of proper
K-moduli of $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic
proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants
([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the
local normalized volume of [Li] and the higher $\Theta$-stable reduction
instead.
