  We study the ground-state entanglement in the quantum Ising model with
nearest neighbor ferromagnetic coupling $J$ and find a sequential increase of
entanglement depth with growing $J$. This entanglement avalanche starts with
two-point entanglement, as measured by the concurrence, and continues via the
three-tangle and four-tangle, until finally, deep in the ferromagnetic phase
for $J=\infty$, arriving at pure $\ell$-partite (GHZ type) entanglement of all
$\ell$ spins. Comparison with the two, three, and four-point correlations
reveals a similar sequence and shows strong ties to the above entanglement
measures for small $J$. However, we also find a partial inversion of the
hierarchy, where the four-point correlation exceeds the three- and two-point
correlations, well before the critical point is reached. Qualitatively similar
behavior is also found for the Bose-Hubbard model, suggesting that this is a
general feature of a quantum phase transition. This should have far reaching
consequences for approximations starting from a mean-field limit.
