  We use difference sets to construct interesting sets of lines in complex
space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in
C^k when k-1 is a prime power. Using semiregular relative difference sets with
parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k.
We show how to construct these difference sets from commutative semifields and
that several known maximal sets of mutually unbiased bases can be obtained in
this way, resolving a conjecture about the monomiality of maximal sets. We also
relate mutually unbiased bases to spin models.
