The dynamics and spread of quantum information in complex many-body systems is presently attracting

a lot of attention across various fields, ranging from cold atom physics via condensed quantum matter

to high energy physics and quantum gravity. This includes questions of how a quantum system thermalizes

and phenomena like many-body interference and localization, more generally non-classicality in many-particle

quantum physics. Here concepts that are based on echoes, i.e. rewinding time, provide a useful way to monitor complex quantum dynamics and its stability. Central to these developments are so-called out-of-time-order correlators (OTOCs) as sensitive probes for chaos and the temporal growth of complexity in interacting systems.

We will address such phenomena for quantum critical and quantum chaotic systems using semiclassical path

integral techniques based on interfering Feynman paths, thereby bridging the classical and quantum many-body world and allowing for deriving random matrix results. These methods enable us to compute echoes and

OTOCs including entanglement and correlation effects. Moreover, on the numerical side we devise a

semiclassical method for Bose-Hubbard systems far-out-of equilibrium that allows us to calculate many-body

quantum interference on time scales far beyond the famous Ehrenfest/scrambling time.