  I first give an overview of the thesis and Matrix Product States (MPS)
representation of quantum spin chains with an improvement on the conventional
notation.
  The rest of this thesis is divided into two parts. The first part is devoted
to eigenvalues of quantum many-body systems (QMBS). I introduce Isotropic
Entanglement, which draws from various tools in random matrix theory and free
probability theory (FPT) to accurately approximate the eigenvalue distribution
of QMBS on a line with generic interactions. Next, I discuss the energy
distribution of one particle hopping random Schr\"odinger operator in 1D from
FPT in context of the Anderson model.
  The second part is devoted to ground states and gap of QMBS. I first give the
necessary background on frustration free (FF) Hamiltonians, real and imaginary
time evolution within MPS representation and a numerical implementation. I then
prove the degeneracy and FF condition for quantum spin chains with generic
local interactions, including corrections to our earlier assertions. I then
summarize my efforts in proving lower bounds for the entanglement of the ground
states, which includes some new results, with the hope that they inspire future
work resulting in solving the conjecture given therein. Next I discuss two
interesting measure zero examples where FF Hamiltonians are carefully
constructed to give unique ground states with high entanglement. One of the
examples (i.e., $d=4$) has not appeared elsewhere. In particular, we calculate
the Schmidt numbers exactly, entanglement entropies and introduce a novel
technique for calculating the gap which may be of independent interest. The
last chapter elaborates on one of the measure zero examples (i.e., $d=3$) which
is the first example of a FF translation-invariant spin-1 chain that has a
unique highly entangled ground state and exhibits signatures of a critical
behavior.
