  We propose a class of generalizations of the geometric entanglement for pure
states by exploiting the matrix product state formalism. This generalization is
completely divested from the notion of separability and can be freely tuned as
a function of the bond dimension to target states which vary in entanglement
complexity. We first demonstrate its value in a toy spin-1 model where, unlike
the conventional geometric entanglement, it successfully identifies the AKLT
ground state. We then investigate the phase diagram of a Haldane chain with
uniaxial and rhombic anisotropies, revealing that the generalized geometric
entanglement can successfully detect all its phases and their entanglement
complexity. Finally we investigate the disordered spin-$1/2$ Heisenberg model,
where we find that differences in generalized geometric entanglements can be
used as lucrative signatures of the ergodic-localized entanglement transition.
