  Recently, Li and Yamazaki proposed a new class of infinite-dimensional
algebras, quiver Yangian, which generalizes the affine Yangian
$\mathfrak{gl}_{1}$. The characteristic feature of the algebra is the action on
BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one
correspondence with the crystal melting models. These algebras can be
bootstrapped from the action on the crystals and have various truncations.
  In this paper, we propose a $q$-deformed version of the quiver Yangian,
referred to as the quiver quantum toroidal algebra (QQTA). We examine some of
the consistency conditions of the algebra. In particular, we show that QQTA is
a Hopf superalgebra with a formal super coproduct, like known quantum toroidal
algebras. QQTA contains an extra central charge $C$. When it is trivial
($C=1$), QQTA has a representation acting on the three-dimensional crystals,
like Li-Yamazaki's quiver Yangian. While we focus on the toric Calabi-Yau
threefolds without compact 4-cycles, our analysis can likely be generalized to
all toric Calabi-Yau threefolds.
