  A systematic and compact treatment of arbitrary $su(2)$ invariant spin-$s$
quantum chains with nearest-neighbour interactions is presented. The
ground-state is derived in terms of matrix product states (MPS). The
fundamental MPS calculations consist of taking products of basic tensors of
rank 3 and contractions thereof. The algebraic $su(2)$ calculations are carried
out completely by making use of Wigner calculus. As an example of application,
the spin-1 bilinear-biquadratic quantum chain is investigated. Various physical
quantities are calculated with high numerical accuracy of up to 7 digits. We
obtain explicit results for the ground-state energy, entanglement entropy,
singlet operator correlations and the string order parameter. We find
interesting crossover phenomena in the correlation lengths.
