Quantum computation suggests a revolution in technology and in cryptography, a completely new perspective on the foundations of theoretical computer science, and a different approach to the study of physical systems.One of the major new developments in quantum computation over the last few years has been the emergence of a new field called ``Quantum Hamiltonian complexity (QHC)'', which sits on the boundary between computational complexity theory and condensed matter physics. This direction investigates computational aspects of physical objects such as ground states and Hamiltonians, using techniques from both physics and theoretical computer science. This direction has already hadan immense impact on both quantum computation and condensed matter physics.This project aims not only to investigate fundamental questions in quantumHamiltonian complexity as it exists today, such as quantum states generation, tensor network descriptions of quantum states, area laws, and the complexity of Hamiltonians, but also to greatly broaden the scope of this new paradigm, into the study of quantum PCP; into new frontiers in quantum algorithms such as quantum walks, adiabatic algorithms and topology and tensor networks related algorithms; as well as into the study of quantum protocols such as coin flipping, quantum interactive proofs and quantum cryptography and their implications on our understanding of quantum entanglement.