  In AdS/CFT, the fine grained entropy of a boundary region is dual to the area
of an extremal surface X in the bulk. It has been proposed that the area of a
certain 'causal surface' C - i.e. the 'causal holographic information' (CHI) -
corresponds to some coarse-grained entropy in the boundary theory. We construct
two kinds of counterexamples that rule out various possible duals, using (1)
vacuum rigidity and (2) thermal quenches. This includes the 'one-point entropy'
proposed by Kelly and Wall, and a large class of related procedures. Also, any
coarse-graining that fixes the geometry of the bulk 'causal wedge' bounded by
C, fails to reproduce CHI. This is in sharp contrast to the holographic
entanglement entropy, where the area of the extremal surface X measures the
same information that is found in the 'entanglement wedge' bounded by X.
