  We establish the existence of several quantum trace maps. The simplest one is
an algebra map between two quantizations of the algebra of regular functions on
the $SL_n$-character variety of a surface $\mathfrak{S}$ equipped with an ideal
triangulation $\lambda$. The first is the (stated) $SL_n$-skein algebra
$\mathscr{S}(\mathfrak{S})$. The second
$\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ is the Fock and Goncharov's
quantization of their $X$-moduli space. The quantum trace is an algebra
homomorphism
$\bar{tr}^X:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{X}}(\mathfrak{S},\lambda)$
where the reduced skein algebra $\overline{\mathscr{S}}(\mathfrak{S})$ is a
quotient of $\mathscr{S}(\mathfrak{S})$. When the quantum parameter is 1, the
quantum trace $\bar{tr}^X$ coincides with the classical Fock-Goncharov
homomorphism. This is a generalization of the Bonahon-Wong quantum trace map
for the case $n=2$. We then define the extended Fock-Goncharov algebra
$\mathcal{X}(\mathfrak{S},\lambda)$ and show that $\bar{tr}^X$ can be lifted to
$tr^X:\mathscr{S}(\mathfrak{S})\to\mathcal{X}(\mathfrak{S},\lambda)$. We show
that both $\bar{tr}^X$ and $tr^X$ are natural with respect to the change of
triangulations. When each connected component of $\mathfrak{S}$ has non-empty
boundary and no interior ideal point, we define a quantization of the
Fock-Goncharov $A$-moduli space $\overline{\mathcal{A}}(\mathfrak{S},\lambda)$
and its extension $\mathcal{A}(\mathfrak{S},\lambda)$. We then show that there
exist quantum traces
$\bar{tr}^A:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{A}}(\mathfrak{S},\lambda)$
and
$tr^A:\mathscr{S}(\mathfrak{S})\hookrightarrow\mathcal{A}(\mathfrak{S},\lambda)$,
where the second map is injective, while the first is injective at least when
$\mathfrak{S}$ is a polygon. They are equivalent to the $X$-versions but have
better algebraic properties.
