Complex systems are ubiquitous in mathematics, biology, engineering, and physics, and the past ten years have witnessed an exponential increase in the literature associated such systems. A shared conceptual framework is becoming apparent among challenges as seemingly different as the following: the search by mathematicians for exact high-order trigonometric identities, the search by engineers for stable control systems, the search by biologists for stable protein structures, and the search by condensed matter physicists for ground states. Recent work has shown that separated product-sum representations provide a powerful and broadly applicable tool for analyzing complex systems. Beylkin and Mohlenkamp provide a good introduction to these representations in their recent preprint "Algorithms for Numerical Analysis in High Dimensions" (*). This talk will review some of the basic ideas of separated product-sum representations, and discuss how our UW Quantum System Engineering (QSE) Group is applying these ideas in polynomial-time simulations of large-scale quantum spin systems. Our QSE Group has found that Beylkin and Mohlenkamp's methods can be readily extended to dynamical systems by a two-fold trick: (1) introduce noise, and (2) convert the noise to an equivalent measurement processes. The second step exploits the same unitary invariance of operator-sum representations that plays a central role in quantum computing theory. The resulting quantum trajectories are readily projected onto low- dimensional manifolds of Beylkin-Mohlenkamp type, where they can be integrated using polynomial-time numerical algorithms. The practical consequence is that a broad class of problems in quantum physics and engineering that were previously thought to be in the (intractable) complexity class EXP can now be solved by algorithms that are in the (much simpler) complexity class NP. The lecture will close with an informal survey of physics problems that might be addressed by these methods. (*) http://amath.colorado.edu/activities/preprints/archive/519.pdf