Genuine multipartite Einstein-Podolsky-Rosen steering

We develop the concept of genuine N-partite Einstein-Podolsky-Rosen (EPR) steering. This nonlocality is the natural multipartite extension of the original EPR paradox. Useful properties emerge that are not guaranteed for genuine multipartite entangled states. In particular, there is a close link with the task of one-sided device-independent quantum secret sharing. We derive inequalities to demonstrate multipartite EPR steering for Greenberger-Horne-Zeilinger (GHZ) and Gaussian continuous variable (CV) states in loophole-free scenarios.