### Abstract

Deciding whether or not a given set of marginal states could have come from a global

physical state is a well-known difficult problem. In this work we consider compatible states from the point of view of quantum correlations. I.e. in case of three parties we often assume that some two-body reduced states are

compatible with a global tri-partite but also carry particular type of correlations. For example in case of qubits we show that if three two-qubit marginals are classical-classical states and they are compatible to some

set of global states then there are always biseparable states in this set. Further we consider situation when two-body (qudit) marginals are separable. In case when these three two-body marginals are compatible to some three-body states we provide necessary conditions for the global state to be genuinly multi-partite entangled, thereby giving a partial answer to a longstanding problem about whether one can deduce with certainty the genuine multipartite nature of the global state from its separable marginals. We provide several three qubit and three qudit examples that fulfill these necessary conditions.