Positivity for quantum cluster algebras from orbifolds

Let $(S,M,U)$ be a marked orbifold with or without punctures and let ${\mathcal{A}}_v$ 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 ${\mathcal{A}}_v$ concerning an arbitrary quantum seed. Consequently, the positivity for the quantum cluster algebra ${\mathcal{A}}_v$ is proved.