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We present a framework for understanding the dynamics of operator size, andbounding the growth of out-of-time-ordered correlators, in models of large-$S$spins. Focusing on the dynamics of a single spin, we show the finiteness of theLyapunov exponent in the large-$S$ limit; our bounds are tighter than the bestknown Lieb-Robinson-type bounds on these systems. We numerically find our upperbound on Lyapunov exponents is within an order of magnitude of numericallycomputed values in classical and quantum kicked top models. Generalizing ourresults to coupled large-$S$ spins on lattices, we show that the butterflyvelocity, which characterizes the spatial speed of quantum informationscrambling, is finite as $S\rightarrow\infty$. We emphasize qualitativedifferences between operator growth in semiclassical large-spin models, andquantum holographic systems including the Sachdev-Ye-Kitaev model.

Quantum operator growth bounds for kicked tops and semiclassical spin chains