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The spectral statistics and entanglement within the eigenstates of genericspin chain Hamiltonians are analysed.  A class of random matrix ensembles is defined which include the most generalnearest-neighbour qubit chain Hamiltonians. For these ensembles, and theirgeneralisations, it is seen that the long chain limiting spectral density is aGaussian and that this convergence holds on the level of individualHamiltonians. The rate of this convergence is numerically seen to be slow.Higher eigenvalue correlation statistics are also considered, the canonicalnearest-neighbour level spacing statistics being numerically observed andlinked with ensemble symmetries. A heuristic argument is given for aconjectured form of the full joint probability density function for theeigenvalues of a wide class of such ensembles. This is numerically verified ina particular case.  For many translationally-invariant nearest-neighbour qubit Hamiltonians it isshown that there exists a complete orthonormal set of eigenstates for which theentanglement present in a generic member, between a fixed length block ofqubits and the rest of the chain, approaches its maximal value as the chainlength increases. Many such Hamiltonians are seen to exhibit a simple spectrumso that their eigenstates are unique up to phase.  The entanglement within the eigenstates contrasts the spectral density forsuch Hamiltonians, which is that seen for a non-interacting chain of qubits.For such non-interacting chains, their always exists a basis of eigenstates forwhich there is no entanglement present.

Quantum spin chains and random matrix theory