  We study the "complexity equals volume" (CV) and "complexity equals action"
(CA) conjectures by examining moments of of time symmetry for $\rm AdS_3$
wormholes having $n$ asymptotic regions and arbitrary (orientable) internal
topology. For either prescription, the complexity relative to $n$ copies of the
$M=0$ BTZ black hole takes the form $\Delta C = \alpha c \chi $, where $c$ is
the central charge and $\chi$ is the Euler character of the bulk time-symmetric
surface. The coefficients $\alpha_V = -4\pi/3$, $\alpha_A = 1/6 $ defined by CV
and CA are independent of both temperature and any moduli controlling the
geometry inside the black hole. Comparing with the known structure of dual CFT
states in the hot wormhole limit, the temperature and moduli independence of
$\alpha_V$, $\alpha_A$ implies that any CFT gate set defining either complexity
cannot be local. In particular, the complexity of an efficient quantum circuit
building local thermofield-double-like entanglement of thermal-sized patches
does not depend on the separation of the patches so entangled. We also comment
on implications of the (positive) sign found for $\alpha_A$, which requires the
associated complexity to decrease when handles are added to our wormhole.
