  We examine a hierarchy of equivalence classes of quasi-random properties of
Boolean Functions. In particular, we prove an equivalence between a number of
properties including balanced influences, spectral discrepancy, local strong
regularity, homomorphism enumerations of colored or weighted graphs and
hypergraphs associated with Boolean functions as well as the $k$th-order strict
avalanche criterion amongst others. We further construct families of
quasi-random boolean functions which exhibit the properties of our equivalence
theorem and separate the levels of our hierarchy.

Combinatorics;Discrete Mathematics