In time-independent quantum systems, entanglement entropy possesses an inherent scaling symmetry that the energy of the system does not have. The symmetry also assures that entropy divergence can be associated with the zero modes. We generalize this symmetry to time-dependent systems all the way from a coupled harmonic oscillator with a time-dependent frequency, to quantum scalar fields with time-dependent mass. We show that such systems have dynamical scaling symmetry that leaves the evolution of various measures of quantum correlations invariant; entanglement entropy, GS fidelity, the Loschmidt echo, and circuit complexity. Using this symmetry, we show that several quantum correlations are related at late times when the system develops instabilities. We then quantify such instabilities in terms of scrambling time and Lyapunov exponents. The delayed onset of exponential decay of the Loschmidt echo is found to be determined by the largest inverted mode in the system. On the other hand, a zero mode retains information about the system for a considerably longer time, finally resulting in a power-law decay of the Loschmidt echo. We extend the analysis to time-dependent massive scalar fields in (1+1)-dimensions and discuss the implications of zero modes and inverted modes occurring in the system at late times. We explicitly show the entropy scaling oscillates between the area-law and volume-law for a scalar field with stable modes or zero modes. We then provide a qualitative discussion of the above effects for scalar fields in cosmological and black hole spacetimes.