It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(fμ,μ) \leqslant C_V \mathrm{Var}_μ(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux \cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Émery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_2^2(fμ,μ)$ and its centralization $W_2^2(f_cμ,μ)$ for $f_c = \frac{|\sqrt{f} - μ(\sqrt{f})|^2}{\mathrm{Var}_μ(\sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(fμ,μ)$ associated to the variance of $\sqrt{f}$ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Götze's characterization of the Talagrand's inequality.