Quantum and stochastic processes

In this dissertation we present results for various quantum and stochastic processes. First, we bound separations between the entangled and classical values for several classes of nonlocal t-player games. We show that for many general classes of such games, the entangled winning probability can not be much higher than the classical one. The proofs use semidefinite-programming techniques and hypergraph norms. Next, we study so-called quasirandom properties of graphs, and extend this to the quantum realm. For a certain class of quantum channels we generalize several results on equivalence between different quasirandom properties, using the non-commutative Grothendieck inequality. We also look at methods for sampling random graphs with power-law degree distributions, using Markov Chain switching methods. Using these samples we present a conjecture on the asymptotic number of triangles in the uniform random graph model. Next we study a class of stochastic processes on graphs that include the discrete Bak-Sneppen process and the contact process. These processes exhibit a phenomenon which we call the Power Light Cone, that has been used in the physics community for a long time but had not yet been proven. We provide this proof, which allows for numerical computations that estimate critical exponents. Finally, we consider a quantum version of Pascal's triangle. When Pascal's triangle is plotted modulo 2, the Sierpinski triangle appears. We prove that when the quantum version of this triangle is plotted modulo 3, a fractal known as the Sierpinski carpet emerges.