  Here we investigate the phase diagram of the SO(n) bilinear-biquadratic
quantum spin chain by studying the global quantum correlations of the ground
state. We consider the cases of n=3,4 and 5 and focus on the geometric
entanglement in the thermodynamic limit. Apart from capturing all the known
phase transitions, our analysis shows a number of novel distinctive behaviors
in the phase diagrams which we conjecture to be general and valid for arbitrary
n. In particular, we provide an intuitive argument in favor of an infinite
entanglement length in the system at a purely-biquadratic point. Our results
are also compared to other methods, such as fidelity diagrams.
