Mid-Term Report Summary - EDEQS (ENTANGLING AND DISENTANGLING EXTENDED QUANTUM SYSTEMS IN AND OUT OF EQUILIBRIUM) Entanglement is a property of quantum matter that has no classical analogue and it recently became a useful concept for characterizing different states of matter as well as non-equilibrium time evolution. This is particularly relevant for those many-body quantum systems in one and two spatial dimensions which exhibit unconventional collective behavior that gives rise to intriguing novel states of matter, such as topological states exhibiting nonabelian statistics and spin-charge separated metals. One main aim of this project is to utilize entanglement to fully classify states of matter in low dimensional systems. This has been already initiated by carrying out a systematic study of the entanglement between two disconnected parts of a one dimensional quantum system quantified by an elusive quantity called entanglement negativity which resisted until now to any analytical computation. A second main aim of this project is to utilize the entanglement to identify the most important features of the the non equilibrium time evolution, with a view to solve exactly the dynamics in strongly interacting integrable models. This allowed to understand which experimental measurable quantities (and why) can be described by standard thermodynamics long time after a perturbation.