  In this article, we study quantum automorphism groups of distance-transitive
graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson
graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We
also give a table with the quantum automorphism groups of all cubic
distance-transitive graphs. Furthermore, with one graph missing, we can now
decide whether or not a distance-regular graph of order $\leq 20$ has quantum
symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum
symmetry. On a final note, we present an example of a pair of graphs with the
same intersection array (the Shrikhande graph and the $4 \times 4$ rook's
graph), where one of them has quantum symmetry and the other one does not.
