Sum of 4 squares decomposition is the well-known result. I'm interested only in negative/non-negative separation with focus on efficiency and large numbers. I'm looking for alternatives or extensions like sum of five or with multipliers.

Any known chance to do better than expected $\log^2(n)$ time (Paul Pollack and Enrique Treviño, "FINDING THE FOUR SQUARES IN LAGRANGE’S THEOREM") with alternatives?