  The existence of higher-spin quantum conserved currents in two dimensions
guarantees quantum integrability. We revisit the question of whether
classically-conserved local higher-spin currents in two-dimensional sigma
models survive quantization. We define an integrability index $\mathcal{I}(J)$
for each spin $J$, with the property that $\mathcal{I}(J)$ is a lower bound on
the number of quantum conserved currents of spin $J$. In particular, a positive
value for the index establishes the existence of quantum conserved currents.
For a general coset model, with or without extra discrete symmetries, we derive
an explicit formula for a generating function that encodes the indices for all
spins. We apply our techniques to the $\mathbb{CP}^{N-1}$ model, the $O(N)$
model, and the flag sigma model $\frac{U(N)}{U(1)^{N}}$. For the $O(N)$ model,
we establish the existence of a spin-6 quantum conserved current, in addition
to the well-known spin-4 current. The indices for the $\mathbb{CP}^{N-1}$ model
for $N>2$ are all non-positive, consistent with the fact that these models are
not integrable. The indices for the flag sigma model $\frac{U(N)}{U(1)^{N}}$
for $N>2$ are all negative. Thus, it is unlikely that the flag sigma models are
integrable.
