  In this short article, we obtained some equivalent formulations of property
$T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the
full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and
the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated
with the trivial unitary corepresentation (that generalize the corresponding
results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type,
we show that $\mathbb{G}$ has property $T$ if and only if every finite
dimensional irreducible $*$-representation of
$C_0^\mathrm{u}(\widehat{\mathbb{G}})$ is an isolated point in the spectrum of
$C_0^\mathrm{u}(\widehat{\mathbb{G}})$ (this also generalizes the corresponding
locally compact group result). In addition, we give a way to construct property
$T$ discrete quantum groups using bicrossed products.
