Geometric influences on quantum Boolean cubes

In this work, we study three problems related to the L₁-influence on quantum Boolean cubes. In the first place, we obtain a dimension free bound for L₁-influence, which implies the quantum L¹-KKL Theorem result obtained by Rouze, Wirth and Zhang. Beyond that, we also obtain a high order quantum Talagrand inequality and quantum L¹-KKL theorem. Lastly, we prove a quantitative relation between the noise stability and L¹-influence. To this end, our technique involves the random restrictions method as well as semigroup theory.