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This paper addresses the problems of quantum spectral curves and 4D limit forthe melting crystal model of 5D SUSY $U(1)$ Yang-Mills theory on$\mathbb{R}^4\times S^1$. The partition function $Z(\mathbf{t})$ deformed by aninfinite number of external potentials is a tau function of the KP hierarchywith respect to the coupling constants $\mathbf{t} = (t_1,t_2,\ldots)$. Asingle-variate specialization $Z(x)$ of $Z(\mathbf{t})$ satisfies a$q$-difference equation representing the quantum spectral curve of the meltingcrystal model. In the limit as the radius $R$ of $S^1$ in $\mathbb{R}^4\timesS^1$ tends to $0$, it turns into a difference equation for a 4D counterpart$Z_{\mathrm{4D}}(X)$ of $Z(x)$. This difference equation reproduces the quantumspectral curve of Gromov-Witten theory of $\mathbb{CP}^1$. $Z_{\mathrm{4D}}(X)$is obtained from $Z(x)$ by letting $R \to 0$ under an $R$-dependenttransformation $x = x(X,R)$ of $x$ to $X$. A similar prescription of 4D limitcan be formulated for $Z(\mathbf{t})$ with an $R$-dependent transformation$\mathbf{t} = \mathbf{t}(\mathbf{T},R)$ of $\mathbf{t}$ to $\mathbf{T} =(T_1,T_2,\ldots)$. This yields a 4D counterpart $Z_{\mathrm{4D}}(\mathbf{T})$of $Z(\mathbf{t})$. $Z_{\mathrm{4D}}(\mathbf{T})$ agrees with a generatingfunction of all-genus Gromov-Witten invariants of $\mathbb{CP}^1$. Fay-typebilinear equations for $Z_{\mathrm{4D}}(\mathbf{T})$ can be derived fromsimilar equations satisfied by $Z(\mathbf{t})$. The bilinear equations implythat $Z_{\mathrm{4D}}(\mathbf{T})$, too, is a tau function of the KP hierarchy.These results are further extended to deformations $Z(\mathbf{t},s)$ and$Z_{\mathrm{4D}}(\mathbf{T},s)$ by a discrete variable $s \in \mathbb{Z}$,which are shown to be tau functions of the 1D Toda hierarchy.

4D limit of melting crystal model and its integrable structure