<p>In this paper we deduce new characterizations for bivariate Bessel-Potential spaces defined on the unit square via B-spline quarklets. For that purpose in a first step we use univariate boundary adapted quarklets to describe univariate Bessel-Potential spaces on intervals. To obtain the bivariate characterizations a recent result of Hansen, Sickel, Yang and Yuan is applied. It yields that each bivariate Bessel-Potential space on a square can be written as an intersection of function spaces which have a tensor product structure. Hence our main result is a characterization of bivariate Bessel-Potential spaces on squares in terms of quarklets that are tensor products of univariate quarklets on intervals.</p>