This paper proposes a set of Lyapunov-type conditions that are suited for stability analysis of large-scale discrete-time systems. A time-wise relaxation of the Lyapunov function decrease condition is employed to derive a set of global and distributed stability conditions. Essentially, these conditions allow to make a trade-off between complexity and conservatism by extending the time-horizon over which the decrease condition should hold. It is shown that for exponentially stable dynamics and any candidate Lyapunov function, there exists a finite time for which the proposed global or distributed stability conditions hold. Hence, it is possible to use functions with a particular structure to make verification of stability scalable for large-scale systems. The developed results are applied to establish stability of a benchmark power systems example.