  We study quadratic residue difference sets, GMW difference sets, and
difference sets arising from monomial hyperovals, all of which are $(2^d-1,
2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of
the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that,
except for a few cases with small $d$, these difference sets are all pairwise
inequivalent. This is accomplished in part by examining their 2-ranks. The
2-ranks of all of these difference sets were previously known, except for those
connected with the Segre and Glynn hyperovals. We determine the 2-ranks of the
difference sets arising from the Segre and Glynn hyperovals, in the following
way. Stickelberger's theorem for Gauss sums is used to reduce the computation
of these 2-ranks to a problem of counting certain cyclic binary strings of
length $d$. This counting problem is then solved combinatorially, with the aid
of the transfer matrix method. We give further applications of the 2-rank
formulas, including the determination of the nonzeros of certain binary cyclic
codes, and a criterion in terms of the trace function to decide for which
$\beta$ in $F_{2^d}^*$ the polynomial $x^6 + x + \beta$ has a zero in
$F_{2^d}$, when $d$ is odd.
