We present a theory of quasiparticle Hall transport in strongly type-II superconductors within their vortex state. We establish the existence of integer quantum spin Hall effect in clean unconventional $d_{x^2-y^2}$ superconductors in the vortex state from a general analysis of the Bogoliubov-de Gennes equation. The spin Hall conductivity $sigma^s_{xy}$ is shown to be quantized in units of $frac{hbar}{8pi}$. This result does not rest on linearization of the BdG equations around Dirac nodes and therefore includes inter-nodal physics in its entirety. In addition, this result holds for a generic inversion-symmetric lattice of vortices as long as the magnetic field $B$ satisfies $H_{c1} ll B ll H_{c2}$. We then derive the Wiedemann-Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of $T to 0$, the thermal Hall conductivity satisfies $kappa_{x y}=frac{4pi^2}{3}(frac{k_B}{hbar})^2 T sigma^s_{xy}$. The transitions between different quantized values of $sigma^s_{xy}$ as well as relation to conventional superconductors are discussed.
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