In this talk, we mainly focus on one problem raised by Professor Quanhua Xu concerning the optimal orders of best constants for the reverse Littlewood-Paley-Stein inequality. First, I will recall the history and recent developments of the Littlewood-Paley-Stein theory. Then I will introduce our results and show our proof by using the Burkholder-Gundy inequality. Our argument is based on the construction of a special symmetric diffusion semigroup associated with any given martingale such that its square function for semigroups is pointwise comparable with its square function for martingales. This reveals deep connections between these two inequalties. Our method also extends to the vector-valued and noncommutative setting.