Fundamental quantum information: correlations, resources, complexity

I will review recent and ongoing progress on three basic subjects in fundamental quantum physics and quantum information. I will first discuss the problem of quantifying quantum correlations beyond entanglement and illustrate some groundbreaking progress on the quest for maximally quantum correlated states. I will then report current advances in the theory and engineering of entangled resources in the post-Gaussian scenario. Finally, I will introduce and discuss novel tools of entanglement theory that may provide hitherto inaccessible insights in the study of complex many-body quantum systems.