  We prove that translationally invariant Hamiltonians of a chain of $n$ qubits
with nearest-neighbour interactions have two seemingly contradictory features.
Firstly in the limit $n\rightarrow\infty$ we show that any translationally
invariant Hamiltonian of a chain of $n$ qubits has an eigenbasis such that
almost all eigenstates have maximal entanglement between fixed-size sub-blocks
of qubits and the rest of the system; in this sense these eigenstates are like
those of completely general Hamiltonians (i.e. Hamiltonians with interactions
of all orders between arbitrary groups of qubits). Secondly in the limit
$n\rightarrow\infty$ we show that any nearest-neighbour Hamiltonian of a chain
of $n$ qubits has a Gaussian density of states; thus as far as the eigenvalues
are concerned the system is like a non-interacting one. The comparison applies
to chains of qubits with translationally invariant nearest-neighbour
interactions, but we show that it is extendible to much more general systems
(both in terms of the local dimension and the geometry of interaction).
Numerical evidence is also presented which suggests that the translational
invariance condition may be dropped in the case of nearest-neighbour chains.
