Geometric influences on quantum Boolean cubes

In this work, we study three problems related to the $L^1$-influence on quantum Boolean cubes. In the first place, we obtain a dimension free bound for $L^1$-influence, which implies the quantum $L^1$-KKL Theorem result obtained by Rouzé, Wirth and Zhang. Beyond that, we also obtain a high order quantum Talagrand inequality and quantum $L^1$-KKL theorem. Lastly, we prove a quantitative relation between the noise stability and $L^1$-influence. To this end, our technique involves the random restrictions method as well as semigroup theory.