  Quantum properties of correlations have a key role in disparate fields of
physics, from quantum information processing, to quantum foundations, to
strongly correlated systems. We tackle a specific aspect of the fundamental
quantum marginal problem: we address the issue of deducing the global
properties of correlations of tripartite quantum states based on the knowledge
of their bipartite reductions, focusing on relating specific properties of
bipartite correlations to global correlation properties. We prove that strictly
classical bipartite correlations may still require global entanglement and that
unentangled---albeit not strictly classical---reductions may require global
genuine multipartite entanglement, rather than simple entanglement. On the
other hand, for three qubits, the strict classicality of the bipartite
reductions rules out the need for genuine multipartite entanglement. Our work
sheds new light on the relation between local and global properties of quantum
states, and on the interplay between classical and quantum properties of
correlations.
