  We compute spin transport in the unitary Fermi gas using the strong-coupling
Luttinger-Ward theory. In the quantum degenerate regime the spin diffusivity
attains a minimum value of $D_s \simeq 1.3 \hbar/m$ approaching the quantum
limit of diffusion for a particle of mass $m$. Conversely, the spin drag rate
reaches a maximum value of $\Gamma_\sd \simeq 1.2 k_B T_F/\hbar$ in terms of
the Fermi temperature $T_F$. The frequency-dependent spin conductivity
$\sigma_s(\omega)$ exhibits a broad Drude peak, with spectral weight
transferred to a universal high-frequency tail $\sigma_s(\omega \to\infty) =
\hbar^{1/2}C/3\pi(m\omega)^{3/2}$ proportional to the Tan contact density $C$.
For the spin susceptibility $\chi_s(T)$ we find no downturn in the normal
phase.
