Variance of Matrix Trace

Question

Given a random variables $X \in \mathbb{R}^n$, and a constant real matrix $Z$, how can the variance given by $Var[Tr(ZXX^T)]$ be calculated? Note that $Z$ is p.s.d and $X$ is $N (0,C)$.

Answer 1

$$ \operatorname{Var}\left[\operatorname{Tr}\left(ZXX^T\right)\right]=\operatorname{Var}\left[\operatorname{Tr}\left(X^TZX\right)\right]=\operatorname{Var}\left[X^TZX\right]\;. $$

Beyond that not much can be said without knowing anything about $X$ and $Z$.