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This thesis is a study of large sets of unit vectors in $\cx^n$ such that theabsolute value of their standard inner products takes on only a small number ofvalues.  We begin with bounds: what is the maximal size of a set of lines with only agiven set of angles? We rederive a series of upper bounds originally due toDelsarte, Goethals and Seidel, but in a novel way using only zonal polynomialsand linear algebra. In the process we get some new results about complex$t$-designs and also some new characterizations of tightness.  Next we consider constructions. We describe some generic constructions usinglinear codes and Cayley graphs, and then move to two specific instances of theproblem: mutually unbiased bases and equiangular lines. Both cases aremotivated by problems in quantum computing, although they have applications indigital communications as well.  Mutually unbiased bases are collections of orthonormal bases with a constantangle between vectors from different bases. We construct some maximal sets inprime-power dimensions, originally due to Calderbank, Cameron, Kantor andSeidel, but again in a novel way using relative difference sets ordistance-regular antipodal covers. We also detail their numerous relations toother combinatorial objects, including symplectic spreads, orthogonaldecompositions of Lie algebras, and spin models. Peripherally, we discussmutually unbiased bases in small dimensions that are not prime powers and inreal vector spaces.  Equiangular lines are collections of vectors with only one angle betweenthem. We use difference sets from finite geometry to construct equiangularlines: these sets do not have maximal size, but they are maximal with respectto having all entries of the same absolute value. We also include some negativeresults about constructions of maximal sets in large dimensions.

Complex Lines with Restricted Angles