  We study thermalization slowing down of a quantum many-body system upon
approach to two distinct integrability limits. Motivated by previous studies of
classical systems, we identify two thermalization time scales: one quantum
Lyapunov time scale is extracted by quantifying operator growth in time in an
appropriately defined basis, while another ergodization time scale is related
to statistics of fluctuations of the time-evolved operator around its mean
value based on the eigenstate thermalization hypothesis. Using a paradigmatic
Quantum Ising chain we find that both timescales diverge upon approach to
integrability. The relative strength of the divergence of the scales depends on
the particular integrable limit. This allows us to define two different
universality classes of quantum thermalization: short- and long-range networks.
