The classical Perron–Frobenius stability theorem for a primitive square stochastic matrix holds central importance across various fields of applied science, including probability, economics, social networks, machine learning, and the Google search engine. Motivated by the significant demand arising from nonlinear consensus problems in multi-agent systems, it is natural to seek a nonlinear analogue of this classical theorem within the framework of cubic stochastic matrices. In this paper, we address the global stability problem for quadratic stochastic operators associated with column primitive cubic doubly stochastic matrices. These results can be regarded as pioneering extensions of the classical Perron–Frobenius stability theorem, broadening its scope from primitive square doubly stochastic matrices to column primitive cubic doubly stochastic matrices.