  Entanglement serves as a foundational pillar in quantum information theory,
delineating the boundary between what is classical and what is quantum. The
common assumption is that higher entanglement corresponds to a greater degree
of `quantumness'. However, this folk belief is challenged by the fact that
classically simulable operations, such as Clifford circuits, can create highly
entangled states. The simulability of these states raises a question: what are
the differences between `low-magic' entanglement, and `high-magic'
entanglement? We answer this question in this work with a rigorous
investigation into the role of magic in entanglement theory. We take an
operational approach to understanding this relationship by studying tasks such
as entanglement estimation, distillation and dilution. This approach reveals
that magic has notable implications for entanglement. Specifically, we find an
operational separation that divides Hilbert space into two distinct regimes:
the entanglement-dominated (ED) phase and magic-dominated (MD) phase. Roughly
speaking, ED states have entanglement that significantly surpasses their magic,
while MD states have magic that dominates their entanglement. The competition
between the two resources in these two phases induces a computational phase
separation between them: there are {sample- and time-efficient} quantum
algorithms for almost any entanglement task on ED states, while these tasks are
{provably computationally intractable} in the MD phase. Our results find
applications in diverse areas such as quantum error correction, many-body
physics, and the study of quantum chaos, providing a unifying framework for
understanding the behavior of quantum systems. We also offer theoretical
explanations for previous numerical observations, highlighting the broad
implications of the ED-MD distinction across various subfields of physics.
