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In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]