A Temperley-Lieb algebra is extracted from the operator structure of a new class of N2 ×N2 braid matrices presented and studied in previous papers and designated as S O(q)(N), S p(q)(N) for the q-deformed orthogonal and symplectic cases, respectively. Spin chain Hamiltonians are derived from such braid matrices and the corresponding chains are studied. Time evolutions of the chains and the possibility of transition of data encoded in the parameters of mixed states from one end to the other are analyzed. The entanglement entropies S(q, N) of eigenstates of the crucial operator, namely, the q-dependent N2 ×N2 projector P0 appearing in the corresponding Hamiltonian are obtained. Study of entanglements generated under the actions of S O(N), S p(N) braid operators, unitarized with imaginary rapidities (spectral parameters) is presented as a perspective.