  Potential anomalies are analysed for the local spin-3 and spin-4 classically
conserved currents in any two-dimensional sigma model on a compact symmetric
space $G/H$, with $G$ and $H$ classical groups. Quantum local conserved charges
are shown to exist in exactly those models which also possess quantum non-local
(Yangian) charges. The possibility of larger sets of quantum local charges is
discussed and shown to be consistent with known S-matrix results and the
behaviour of the corresponding Yangian representations.
