  The resource theory of asymmetry is a framework for classifying and
quantifying the symmetry-breaking properties of both states and operations
relative to a given symmetry. In the special case where the symmetry is the set
of translations generated by a fixed observable, asymmetry can be interpreted
as coherence relative to the observable eigenbasis, and the resource theory of
asymmetry provides a framework to study this notion of coherence. We here show
that this notion of coherence naturally arises in the context of quantum speed
limits. Indeed, the very concept of speed of evolution, i.e., the inverse of
the minimum time it takes the system to evolve to another (partially)
distinguishable state, is a measure of asymmetry relative to the time
translations generated by the system Hamiltonian. Furthermore, the celebrated
Mandelstam-Tamm and Margolus-Levitin speed limits can be interpreted as upper
bounds on this measure of asymmetry by functions which are themselves measures
of asymmetry in the special case of pure states. Using measures of asymmetry
that are not restricted to pure states, such as the Wigner-Yanase skew
information, we obtain extensions of the Mandelstam-Tamm bound which are
significantly tighter in the case of mixed states. We also clarify some
confusions in the literature about coherence and asymmetry, and show that
measures of coherence are a proper subset of measures of asymmetry.
