Computing the time evolution of closed ergodic systems is a formidable task because all the exponentially large Hilbert space of the system is in principle reachable from a generic initial state. Even if we start from a product state, thermalization involves build up of highly non-local quantum correlations or entanglement in the system, which cannot be captured by modern methods such as t-DMRG or TEBD. But do we really need to accurately keep track of non-local quantum correlations? We know that at long times a thermalizing system should be characterized by an emergent classical behavior dominated by hydrodynamic transport of conserved quantities. In this talk I will discuss how this quantum to classical crossover that occurs during the time evolution can be described and utilized to formulate a systematic approximation scheme for the dynamics of many-body quantum systems. I will use the new scheme to study infinite temperature transport and thermalization in spin chains. I will also comment on the possibility of quantifying quantum chaos in such systems.