  We study emerging notions of quantum correlations in compound systems. Based
on different definitions of quantumness in individual subsystems, we
investigate how they extend to the joint description of a composite system.
Especially, we study the bipartite case and the connection of two typically
applied and distinctively different concepts of nonclassicality in quantum
optics and quantum information. Our investigation includes the representation
of correlated states in terms of quasiprobability matrices, a comparative study
of joint and conditional quantum correlations, and an entanglement
characterization. It is, for example, shown that our composition approach
always includes entanglement as one form of quantum correlations. Yet, other
forms of quantum correlations can also occur without entanglement. Finally, we
give an outlook towards multimode systems and temporal correlations.
