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Building upon the results of [R. Augusiak et al., Phys. Rev. Lett. 115,030404 (2015)] we develop a general approach to the generation of genuinelyentangled multipartite states of any number of parties from genuinely entangledstates of a fixed number of parties, in particular, the bipartite entangledones. In our approach, certain isometries whose output subspaces are eithersymmetric or genuinely entangled in some multipartite Hilbert spaces areapplied to local subsystems of bipartite entangled or multipartite genuinelyentangled quantum states. To prove that entanglement of the resulting states isindeed genuine we then introduce novel criteria allowing to decide itefficiently. The construction is then exploited to provide examples ofmultipartite states that are genuinely entangled but not genuinely nonlocal,giving further illustration for the inequivalence between entanglement andnonlocality in the multiparticle scenario. It is also shown how to constructgenuinely entangled states which are unsteerable across certain bipartite cuts.

Constructions of genuinely entangled multipartite states with applications to local hidden variables (LHV) and states (LHS) models