It is known that a quadratic transportation-information inequality $\mathrm{W_2I}$ interpolates between the Talagrand's inequality $\mathrm{W_2H}$ and the log-Sobolev inequality (LSI for short). The aim of the present paper is threefold: (1) To prove the equivalence of $\mathrm{W_2I}$ and the Lyapunov condition, which gives a new characterization inspired by Cattiaux-Guillin-Wu [8]. (2) To prove the stability of $\mathrm{W_2I}$ under bounded perturbations, which gives a transference principle in the sense of Holley-Stroock. (3) To prove $\mathrm{W_2H}$ through a restricted $\mathrm{W_2I}$, which gives a counterpart of the restricted LSI presented by Gozlan-Roberto-Samson [15].