  We consider the Topological String/Spectral theory duality on toric
Calabi-Yau threefolds obtained from the resolution of the cone over the
$Y^{N,0}$ singularity. Assuming Kyiv formula, we demonstrate this duality in a
special regime thanks to an underlying connection between spectral determinants
of quantum mirror curves and the non-autonomous (q)-Toda system. We further
exploit this link to connect small and large time expansions in Toda equations.
In particular we provide an explicit expression for their tau functions at
large time in terms of a strong coupling version of irregular $W_N$ conformal
blocks at $c=N-1$. These are related to a special class of multi-cut matrix
models which describe the strong coupling regime of four dimensional,
$\mathcal{N}=2$ $SU(N)$ super Yang-Mills.
