  We investigate the tracial states and $\mathbb{G}$-invariant states on the
reduced $C^*$-algebra of a discrete quantum group $\mathbb{G}$, where we denote
its dual compact quantum group by $\hat{\mathbb{G}}$. Certain key results in
this work include that $C_r(\hat{\mathbb{G}})$ admits a tracial state if and
only if it admits a $\mathbb{G}$-invariant state, and that the quantum subgroup
associated with the Furstenberg boundary of $\mathbb{G}$, denoted
$\hat{\mathbb{G}}_F$, is a quantum subgroup of every quantum subgroup of
$\hat{\mathbb{G}}$ of Kac type that has a coamenable quotient. The latter
result establishes that $\hat{\mathbb{G}}_F$ is the "coamenable coradical" of
$\hat{\mathbb{G}}$ whenever its quotient is coamenable and $\hat{\mathbb{G}}$
is of Kac type. The existence of such an object was asked for by Kalantar,
Kasprzak, Skalski, and Vergnioux. With these results, given the assumption that
$\hat{\mathbb{G}}/\hat{\mathbb{G}}_F$ is coamenable, we prove that
$C_r(\hat{\mathbb{G}})$ admits a tracial state if and only if
$\hat{\mathbb{G}}_F$ is of Kac type, and a characterization of uniqueness of
trace in terms of whether or not $\hat{\mathbb{G}}_F$ is equal to the canonical
Kac quotient of $C_r(\hat{\mathbb{G}})$. We also obtain that
$C_r(\hat{\mathbb{G}})$ is nuclear and has a tracial state if and only if
$\mathbb{G}$ is amenable, which resolves an open problem due to C.-K Ng and
Viselter, and Crann, in the discrete case.
