  The stranglehold of low temperatures on fascinating quantum phenomena in
one-dimensional quantum magnets has been challenged recently by the discovery
of anomalous spin transport at high temperatures. Whereas both regimes have
been investigated separately, no study has attempted to reconcile them. For
instance, the paradigmatic quantum Heisenberg spin-$1/2$ chain falls at low
temperature within the Tomonaga-Luttinger liquid framework, while its
high-temperature dynamics is superdiffusive and relates to the
Kardar-Parisi-Zhang universality class in $1+1$ dimensions. This Letter aims at
reconciling the two regimes. Building on large-scale matrix product state
simulations, we find that they are connected by a temperature-dependent
spatiotemporal crossover. As the temperature $T$ is reduced, we show that the
onset of superdiffusion takes place at longer length and timescales $\propto
1/T$. This prediction has direct consequences for experiments including nuclear
magnetic resonance: it is consistent with earlier measurements on the nearly
ideal Heisenberg $S=1/2$ chain compound Sr$_2$CuO$_3$, yet calls for new and
dedicated experiments.
