  We study tight projective 2-designs in three different settings. In the
complex setting, Zauner's conjecture predicts the existence of a tight
projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman
recently proposed an approach to make quantitative progress on this conjecture
in terms of the entanglement breaking rank of a certain quantum channel. We
show that this quantity is equal to the size of the smallest weighted
projective 2-design. Next, in the finite field setting, we introduce a notion
of projective 2-designs, we characterize when such projective 2-designs are
tight, and we provide a construction of such objects. Finally, in the
quaternionic setting, we show that every tight projective 2-design for H^d
determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of
R^d(2d+1) of dimension 3.
