Multi-posets　as　the　generalization　of　quantum　B-algebras　and　partially　ordered　semigroups　etc.　were　firstly　introduced　and　studied　by　Rump.　In　this　paper,　we　shall　continue　his　work　and　investigate　the　completions　of　multi-posets　and　quantum　B-algebras.　Firstly,　we　introduce　the　notions　of　the　(precoherent)　completions　of　a　multi-poset　M　and　the　(algebraic)　consistent　quantic　nuclei　on　the　power-set　beta(m(f))　of　the　free　semigroup　of　M,　and　prove　that　the　(precoherent)　completions　of　a　multi-poset　M　is　completely　determined　by　the　(algebraic)　consistent　quantic　nuclei　on　beta(m(f))　up　to　isomorphism.　Moreover,　we　give　the　concrete　forms　of　the　least　and　the　largest　standard　(precoherent)　completions　and　obtain　the　free　quantale　over　a　multi-poset　in　the　category　of　multi-posets.　Finally,　when　the　completions　and　consistent　quantic　nuclei　of　multi-posets　are　applied　to　the　cases　of　quantum　B-algebras　and　partially　ordered　semigroups,　several　analogous　results　can　also　be　obtained.　(C)　2019　Elsevier　B.V.　All　rights　reserved.