The author analyzes the equivalence of quadrangulations on the sphere \(S^2\). By definition, a quadrangulation on a closed surface \(F^2\) is a map of a simple graph (with no loops and no multiple edges) embedded on \(F^2\) such that each face is quadrilateral. Constructing a particular operation, the diagonal slide, the author proves that any two quadrangulations \(G_1\) and \(G_2\) with \(|V(G_1)|=|V(G_2)|=n\) on the sphere can be transformed into each other, up to homeomorphism, by at most \(6n-32\) diagonal slides and rotations if \(n\geq 6\).