  The spectral statistics and entanglement within the eigenstates of generic
spin chain Hamiltonians are analysed.
  A class of random matrix ensembles is defined which include the most general
nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their
generalisations, it is seen that the long chain limiting spectral density is a
Gaussian and that this convergence holds on the level of individual
Hamiltonians. The rate of this convergence is numerically seen to be slow.
Higher eigenvalue correlation statistics are also considered, the canonical
nearest-neighbour level spacing statistics being numerically observed and
linked with ensemble symmetries. A heuristic argument is given for a
conjectured form of the full joint probability density function for the
eigenvalues of a wide class of such ensembles. This is numerically verified in
a particular case.
  For many translationally-invariant nearest-neighbour qubit Hamiltonians it is
shown that there exists a complete orthonormal set of eigenstates for which the
entanglement present in a generic member, between a fixed length block of
qubits and the rest of the chain, approaches its maximal value as the chain
length increases. Many such Hamiltonians are seen to exhibit a simple spectrum
so that their eigenstates are unique up to phase.
  The entanglement within the eigenstates contrasts the spectral density for
such Hamiltonians, which is that seen for a non-interacting chain of qubits.
For such non-interacting chains, their always exists a basis of eigenstates for
which there is no entanglement present.
