  We advance the characterization of complexity in quantum many-body systems by
examining $W$-states embedded in a spin chain. Such states show an amount of
non-stabilizerness or "magic" (measured as the Stabilizer R\'enyi Entropy
-SRE-) that grows logarithmic with the number of qubits/spins. We focus on
systems whose Hamiltonian admits a classical point with an extensive
degeneracy. Near these points, a Clifford circuit can convert the ground state
into a $W$-state, while in the rest of the phase to which the classic point
belongs, it is dressed with local quantum correlations. Topological frustrated
quantum spin-chains host phases with the desired phenomenology, and we show
that their ground state's SRE is the sum of that of the $W$-states plus an
extensive local contribution. Our work reveals that $W$-states/frustrated
ground states display a non-local degree of complexity that can be harvested as
a quantum resource and has no counterpart in GHZ states/non-frustrated systems.
