The Family-Vicsek scaling is a fundamental framework for understanding surface growth in nonequilibrium classical systems, providing a universal description of temporal surface roughness evolution. Whether such scaling extends to strongly correlated quantum many-body systems—where dynamics are governed by unitarity, conservation laws, and quantum statistics—remains an open question. Here, we investigate the infinite-temperature dynamics of one-dimensional $SU\left(N\right)$ spin chains, focusing on the well-known $SU\left(2\right)$ XXZ model and the $SU\left(3\right)$ Izergin-Korepin model. We compute the quantum analog of classical surface roughness using the second cumulant of spin fluctuations and demonstrate Family-Vicsek scaling with respect to time and subsystem size. Crucially, we show that the scaling behavior is controlled by both integrability and non-Abelian global symmetries. We identify distinct transport regimes characterized by the dynamical exponent $z$: (i) ballistic transport with $z=1$, (ii) superdiffusive transport with the Kardar-Parisi-Zhang exponent $z=3/2$, and (iii) diffusive transport with the Edwards-Wilkinson exponent $z=2$. Notably, breaking integrability always drives the system into the diffusive regime. Our results demonstrate that Family-Vicsek scaling extends beyond classical systems, and emerges across quantum many-body models with $SU\left(N\right)$ symmetry.

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• Quantum quench
• 1-dimensional spin chains
• Kardar–Parisi–Zhang equation
• SU(N) symmetries
• XYZ model