Multipartite entanglement is one of the hallmarks of quantum mechanics and is central to quantum information processing. In this work we show that Concentratable Entanglement (CE), an operationally motivated entanglement measure, induces a hierarchy upon pure states from which different entanglement structures can be experimentally certified. In particular, we find that nearly all genuine multipartite entangled states can be verified through the CE. Interestingly, GHZ states prove to be far from maximally entangled according to this measure. Instead we find the exact maximal value and corresponding states for up to 18 qubits and show that these correspond to extremal quantum error correcting codes. The latter allows us to unravel a deep connection between CE and coding theory. Finally, our results also offer an alternative proof, on up to 31 qubits, that absolutely maximally entangled states do not exist.

A powerful operational paradigm for distributed quantum information processing involves manipulating pre-shared entanglement by local operations and classical communication (LOCC). The LOCC round complexity of a given task describes how many rounds of classical communication are needed to complete the task. Despite some results separating one-round versus two-round protocols, very little is known about higher round complexities. In this paper, we revisit the task of one-shot random-party entanglement distillation as a way to highlight some interesting features of LOCC round complexity. We first show that for random-party distillation in three qubits, the number of communication rounds needed in an optimal protocol depends on the entanglement measure used; for the same fixed state some entanglement measures need only two rounds to maximize whereas others need an unbounded number of rounds. In doing so, we construct a family of LOCC instruments that require an unbounded number of rounds to implement. We then prove explicit tight lower bounds on the LOCC round number as a function of distillation success probability. Our calculations show that the original W-state random distillation protocol by Fortescue and Lo is essentially optimal in terms of round complexity.