  We provide two simple universality criteria. Our first criterion states that
$\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms
a $\delta$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\geq3$.
Our second universality criterion says that $\mathcal{S}\subset G_d$ is
universal if and only if the centralizer of
$\mathcal{S}^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in
\mathcal{S}\}$ is equal to the centralizer of $G_d^{t(d),t(d)}=\{U^{\otimes
t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in G_d\}$, where $t(2)=3$, and $t(d)=2$
for $d\geq 3$. The equality of the centralizers can be verified by comparing
their dimensions.
