Fix $K$ a $p$-adic field and denote by $G_K$ its absolute Galois group. Let $K_{\infty }$ be the extension of $K$ obtained by adding $p^n$-th roots of a fixed uniformizer, and $G_{\infty }\subset G_K$ its absolute Galois group. In this article, we define a class of $p$-adic torsion representations of $G_{\infty }$, calledquasi-semi-stable. We prove that these representations are “explicitly” described by a certain category of linear algebraic objects. The results of this note should be considered as a first step in the understanding of the structure of quotient...