  We define circuits given by unitary representations of Lorentzian conformal
field theory in 3 and 4 dimensions. Our circuits start from a spinning primary
state, allowing us to generalize formulas for the circuit complexity obtained
from circuits starting from scalar primary states. These results are nicely
reproduced in terms of the geometry of coadjoint orbits of the conformal group.
In contrast to the complexity geometry obtained from scalar primary states, the
geometry is more complicated and the existence of conjugate points, signaling
the saturation of complexity, remains open.
