  We study the evolution of holographic complexity of pure and mixed states in
$1+1$-dimensional conformal field theory following a local quench using both
the "complexity equals volume" (CV) and the "complexity equals action" (CA)
conjectures. We compare the complexity evolution to the evolution of
entanglement entropy and entanglement density, discuss the Lloyd computational
bound and demonstrate its saturation in certain regimes. We argue that the
conjectured holographic complexities exhibit some non-trivial features
indicating that they capture important properties of what is expected to be
effective (or physical) complexity.
