  We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum
cluster algebra of the Kronecker quiver which are quantum analogues of the
canonical basis, semicanonical basis and dual semicanonical basis of the
cluster algebra of the Kronecker quiver in the sense of
\cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we
prove the positivity of the elements in these bases.
