The quasiseparable structure can be efficiently exploited – in order to speed up computations and storage – by using hierachical representations. This theme has raised much interest in the last two decades, in particular in the context of elliptic partial differential equations, matrix functions and Sylvester equations. Our aim is to provide a framework for the analysis of the off-diagonal singular values, focusing on the structures that can ensure an exponential decay of them. We see how this structures appears in the computation of matrix functions and in the execution of the cyclic reduction paving the way for a theoretical understanding of the preservation of the quasiseparable rank. We provide family of bounds that highlight which parameters of the model can affect the rate of decay. Finally, we show how to practically exploit the low quasiseparable rank using hierarchical matrices.