  We propose a method which we call "Isotropic Entanglement" (IE), that
predicts the eigenvalue distribution of quantum many body (spin) systems (QMBS)
with generic interactions. We interpolate between two known approximations by
matching fourth moments. Though, such problems can be QMA-complete, our
examples show that IE provides an accurate picture of the spectra well beyond
what one expects from the first four moments alone. We further show that the
interpolation is universal, i.e., independent of the choice of local terms.
