By contrast, the scrambling of quantum information is ballistic and hence can be characterized by a "butterfly" velocity. One way of characterizing the propagation of quantum information is to study out-of-time ordered (OTO) correlation functions, which are unconventional correlation functions with time arguments that are not ordered from early to late times. Using matrix-product-operator based numerical simulations, we compute both time-ordered and OTO correlators at high temperatures in a one-dimensional Bose-Hubbard model, where well defined quasi-particles cease to exist. For the time-ordered correlators of conserved quantities, we demonstrate an emergent hydrodynamic description at late times and compute the associated diffusion constant. Unconventional OTO correlators exhibit by contrast the fast ballistic information propagation, even in the high temperature regime. Furthermore, we discuss how these fundamental properties of nonequilibrium quantum dynamics can be characterized in experiments.