  We prove that synthetic lower Ricci bounds for metric measure spaces -- both
in the sense of Bakry-\'Emery and in the sense of Lott-Sturm-Villani -- can be
characterized by various functional inequalities including local Poincar\'e
inequalities, local logarithmic Sobolev inequalities, dimension independent
Harnack inequality, and logarithmic Harnack inequality.
  More generally, these equivalences will be proven in the setting of
time-dependent metric measure spaces and will provide a characterization of
super-Ricci flows of metric measure spaces.
