  The paper is concerned with the extension of the classical study of
probability measures on a compact group which are square roots of the Haar
measure, due to Diaconis and Shahshahani, to the context of compact quantum
groups. We provide a simple characterisation for compact quantum groups which
admit no non-trivial square roots of the Haar state in terms of their
corepresentation theory. In particular it is shown that such compact quantum
groups are necessarily of Kac type and their subalgebras generated by the
coefficients of a fixed two-dimensional irreducible corepresentation are
isomorphic (as finite quantum groups) to the algebra of functions on the group
of unit quaternions. An example of a quantum group whose Haar state admits no
nontrivial square root and which is neither commutative nor cocommutative is
given.
