The quantum marginal problem asks whether a set of given density matrices are consistent, i.e., whether they can be the reduced density matrices of a global quantum state. Not many non-trivial analytic necessary (or sufficient) conditions are known for the problem in general. We propose a method to detect consistency of overlapping quantum marginals by considering the separability of some derived states. Our method works well for the k-symmetric extension problem in general, and for the general overlapping marginal problems in some cases. Our work is, in some sense, the converse to the well-known k-symmetric extension criterion for separability.The quantum marginal problem, also known as the consistency problem, asks for the conditions under which there exists an N -particle density matrix ρ N whose reduced density matrices (quantum marginals) on the subsets of particles S i ⊂ {1, 2, . . . , N } equal to the given density matrices ρ Si for all i [1]. The related problem in fermionic (bosonic) systems is the so-called N -representability problem. It asks whether a k-fermionic (bosonic) density matrix is the reduced density matrix of some N -fermion (boson) state ρ N . The Nrepresentability problem inherits a long history in quantum chemistry [2,3].The quantum marginal problem and the N -representability problem are in general very difficult. They were shown to be the complete problems of the complexity class QMA, even for the relatively simple case where the given marginals are two-particle states [4][5][6]. In other words, even with the help of a quantum computer, it is very unlikely that the quantum marginal problems can be solved efficiently in the worst case. In this sense, the best hope to have simple analytic conditions for the quantum marginal problem is to find either necessary or sufficient conditions in certain special cases.When the given marginals are states of non-overlapping subsets of particles, and one is interested in a global pure state consistent with the given marginals, both the quantum marginal problem and the N -representability problem were solved [1, 7-11]. However, not much is known for the general problem with overlapping subsystems. For the tripartite case of particles A, B, C, the strong subadditivity of von Neumann entropy enforces non-trivial necessary conditions for the consistency of ρ AB and ρ AC such as S(AB) + S(AC) ≥ S(B) + S(C) [12]. In a similar spirit, certain quantitative monogamy of entanglement type of results (see e.g. [13]) also put non-trivial necessary conditions. Necessary and sufficient conditions are generally not known, except in very few special situations [12,14,15] when N is small.In this work, we propose a simple but powerful analytic necessary condition for arguably the simplest overlapping quantum marginal problem, known as the k-symmetric extension problem. That is, we will consider quantum marginal problems of k + 1 particles A, B 1 , B 2 , . . . , B k for a given density matrix ρ AB , and require that there is a global quantum state ρ AB1B2···B k whose m...