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Homogeneity across lineages is a common assumption in phylogenetics according to which nucleotide substitution rates remain constant in time and do not depend on lineages. This is a simplifying hypothesis which is often adopted to make the process of sequence evolution more tractable. However, its validity has been explored and put into question in several papers. On the other hand, dealing successfully with the general case (heterogeneity across lineages) is one of the key features of phylogenetic reconstruction methods based on algebraic tools. The goal of this paper is twofold. First, we present a new weighting system for quartets (ASAQ) based on algebraic and semi-algebraic tools, thus specially indicated to deal with data evolving under heterogeneus rates. This method combines the weights two previous methods by means of a test based on the positivity of the branch length estimated with the paralinear distance. ASAQ is statistically consistent when applied to GM data, considers rate and base composition heterogeneity among lineages and does not assume stationarity nor time-reversibility. Second, we test and compare the performance of several quartet-based methods for phylogenetic tree reconstruction (namely, Quartet Puzzling, Weight Optimization and Wilson's method) in combination with ASAQ weights and other weights based on algebraic and semi-algebraic methods or on the paralinear distance. These tests are applied to both simulated and real data and support Weight Optimization with ASAQ weights as a reliable and successful reconstruction method.