Abstract: I will present some recent results on entanglement detection and quantification with collective measurements in many-body ensembles. First I will give an introduction into entanglement and the idea of 'spin squeezing', which was introduced in the context of metrology, and explain the relation between the two concepts. I will show how the original spin squeezing approach  can be generalized in several respects and how it allows to quantify multipartite entanglement in different types of experimentally-controlled many-body systems, such as cold atomic clouds or solid-state magnetic materials. These entanglement witnesses are based on variances of collective operators, which can be extracted from simple averaged two-body correlation functions, which is the reason why they find widespread application in many-body systems, where higher-order correlation functions of more complex measurements are generally very challenging. In particular I will present particular examples of criteria that have been recently applied to quantify entanglement in experiments with cold or ultracold atomic gases. Similarly, I will present criteria tailored to detect bipartite entanglement in a many-body state split in two spatially separated sub-ensembles.

In the final part, I will focus on the quantification of entanglement by means of entanglement monotones with similar methods, and again, as a concrete application, I will show the results of applying this method with experimental data of a spin-squeezed Bose-Einstein condensates of ∼500 atoms.