  We study a quantum (non-commutative) representation of the affine Weyl group
mainly of type $E_8^{(1)}$, where the representation is given by birational
actions on two variables $x,y$ with $q$-commutation relations. Using the tau
variables, we also construct quantum "fundamental" polynomials $F(x,y)$ which
completely control the Weyl group actions. The geometric properties of the
polynomials $F(x,y)$ for the commutative case is lifted distinctively in the
quantum case to certain singularity structures as the $q$-difference operators.
This property is further utilized as the characterization of the quantum
polynomials $F(x,y)$. As an application, the quantum curve associated with
topological strings proposed recently by the first named author is rederived by
the Weyl group symmetry. The cases of type $D_5^{(1)}$, $E_6^{(1)}$,
$E_7^{(1)}$ are also discussed.
