I will introduce an ordering problem about a simple loop on a triangulated surface. Then I will explain Allegretti-Kim`s construction of regular functions on quantum Teichmüller space of a punctured surface, using Bonahon-Wong`s quantum trace map. Finally, I will briefly explain how the ordering problem yields the positivity of the coefficients of these regular functions, when written as Laurent polynomials in quantum cluster X variables. This talk is based on the joint paper with S. Cho, H. Kim, and D. Oh (arXiv:1710.06217).