Unique ring signatures (URS) were introduced by Franklin and Zhang (FC 2012) as a unification of linkable and traceable ring signatures. In URS, each member within a ring can only produce, on behalf of the ring, at most one signature for a message. Applications of URS potentially are e���voting systems and e���token systems. In blockchain technology, URS have been implemented for mixing contract. However, existing URS schemes are based on the Discrete Logarithm Problem, which is insecure in the post-quantum setting. In this paper, we design a new lattice-based URS scheme where the signature size is logarithmic in number of ring members. The proposed URS exploits a Merkle tree-based accumulator as building block in the lattice setting. Our scheme is secure under the Short Integer Solution and Learning With Rounding assumptions in the random oracle model.