Holographic complexity is nonlocal

We study the complexity equals volume (CV) and complexity equals action (CA) conjectures by examining moments of of time symmetry for 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 timesymmetric 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 thermo field-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.