  We use the functional renormalization approach for quantum spin systems
developed by Krieg and Kopietz [Phys. Rev. B $\mathbf{99}$, 060403(R) (2019)]
to calculate the spin-spin correlation function $G (\boldsymbol{k}, \omega )$
of quantum Heisenberg magnets at infinite temperature. For small wavevectors
$\boldsymbol{k} $ and frequencies $\omega$ we find that $G ( \boldsymbol{k},
\omega )$ assumes the diffusive form predicted by hydrodynamics. Our result for
the spin-diffusion coefficient ${\cal{D}}$ is somewhat smaller than previous
theoretical predictions based on the extrapolation of the short-time expansion,
but is still about $30 \%$ larger than the measured high-temperature value of
${\cal{D}}$ in the Heisenberg ferromagnet Rb$_2$CuBr$_4\cdot$2H$_2$O. In
reduced dimensions $d \leq 2$ we find superdiffusion characterized by a
frequency-dependent complex spin-diffusion coefficient ${\cal{D}} ( \omega )$
which diverges logarithmically in $d=2$, and as a power-law ${\cal{D}} ( \omega
) \propto \omega^{-1/3}$ in $d=1$. Our result in one dimension implies scaling
with dynamical exponent $z =3/2$, in agreement with recent calculations for
integrable spin chains. Our approach is not restricted to the hydrodynamic
regime and allows us to calculate the dynamic structure factor $S (
\boldsymbol{k} , \omega )$ for all wavevectors. We show how the
short-wavelength behavior of $S ( \boldsymbol{k}, \omega )$ at high
temperatures reflects the relative sign and strength of competing exchange
interactions.
