  The stated skein algebra of a punctured bordered surface (or equivalently, a
marked surface) is a generalization of the well-known Kauffman bracket skein
algebra of unmarked surfaces and can be considered as an extension of the
quantum special linear group $\mathcal{O}_{q^2}(SL_2)$ from a bigon to general
surfaces. We show that the stated skein algebra of a punctured bordered surface
with non-empty boundary can be embedded into quantum tori in two different
ways. The first embedding can be considered as a quantization of the map
expressing the trace of a closed curve in terms of the shear coordinates of the
enhanced Teichm\"uller space, and is a lift of Bonahon-Wong's quantum trace
map. The second embedding can be considered as a quantization of the map
expresses the trace of a closed curve in terms of the lambda length coordinates
of the decorated Teichm\"uller space, and is an extension of Muller's quantum
trace map. We explain the relation between the two quantum trace maps. We also
show that the quantum cluster algebra of Muller is equal to a reduced version
of the stated skein algebra. As applications we show that the stated skein
algebra is an orderly finitely generated Noetherian domain and calculate its
Gelfand-Kirillov dimension.
