It is [1]. Quantum computers, using Shor's algorithm, polynomially break any specialization of the abelian hidden subgroup problem; see [2] for a fairly complete list.

Whatever reason to prefer elliptic curves over integer factorization or discrete log-based schemes must be classical.

[1] http://arxiv.org/abs/quantph/0301141

[2] http://pqcrypto.org/quantum.html