Positivity for quantum cluster algebras from orbifolds

Let (S, M, U) be a marked orbifold with or without punctures and let Aᵥ be a quantum cluster algebra from (S, M, U) with arbitrary coefficients and quantization. We provide combinatorial formulas for quantum Laurent expansion of quantum cluster variables of Aᵥ concerning an arbitrary quantum seed. Consequently, the positivity for the quantum cluster algebra Aᵥ is proved.