It has been known since the work of Bell that entanglement allows for the generation of super-strong correlation between spatially isolated parties. The study of entanglement in many-body systems, however, has revealed important constraints on these correlations as the number of systems increases. The exploitation of the key such constraint, entanglement monogamy, is behind a number of recent results in the study of entanglement in ground states of gapped Hamiltonians as well as the quantum PCP conjecture.

All known results seeking to exploit monogamy to its fullest extent, such as de Finetti-type theorems, only provide strong bounds as the number of systems under study increases. Indeed it seems natural that monogamy dictates an asymptotic weakening of the strength of entanglement as it is shared between more and more parties. 

In this talk we will ask the question whether there is an analogue of monogamy that already manifests itself strongly when only a constant number of systems are available. We will give a positive answer to this question by introducing tools from complexity theory, including the celebrated BLR linearity test. We will show that the constraints imposed by this test lead to a new formulation of asymptotic monogamy-like constraints on entanglement in tripartite systems. As an application, we will use this result towards the resolution of a long-standing open question in quantum complexity theory.