Quantum complexity: from quantum information to quantum matter and quantum gravity

I will review at a general level some recent developments that have occurred in the so-called second quantum revolution. The latter started about three decades ago with the discovery that fundamental but counterintuitive features of quantum mechanics such as quantum entanglement and quantum nonlocality are indeed real physical resources. As such, they can be exploited to realize algorithmic, computational, engineering, and technological tasks with schemes, efficiencies and speedups otherwise impossible by classical means. The central quantity in quantum information theory, the von Neumann entanglement entropy, along with its statistical properties and scaling features (the "area law" and its violations), turns out to play a key role not only in quantum computation and communication but as well in many other apparently detached areas of research, ranging from estimation, sensing and metrology to machine learning, simulation and artificial intelligence, and from condensed matter physics to quantum gravity.