  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 certified. In particular, we find that nearly all genuine
multipartite entangled states can be verified through CE. In the process we
find the exact maximal value of CE 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.
