We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity on the congested clique with about $N^{1/2}$ nodes, where $N$ is the input size. We show that the average time complexity of the local computation performed at a clique node (in terms of the size of the data received by the node) in such protocols has to be substantially larger than the time complexity of the given problem.