  In this paper, we ask the question: How well can Coulomb blockade experiments
correctly identify and distinguish between different topological orders in
quantum Hall states? We definitively find the answer to be: Quite poorly. In
particular, we write the general expression for the spacing of resonance peaks
in a simple form that explicitly displays its dependence on the conformal
scaling dimensions of the systems' edge modes. This form makes transparent the
general argument that the Coulomb blockade peak spacings do not provide a
strongly indicative signature of the topological order of the system, since it
is only weakly related to the braiding statistics. We bolster this general
argument with examples for all the most physically relevant non-Abelian
candidate states, demonstrating that they have Coulomb blockade doppelgangers
-- candidate states at the same filling fraction with identical Coulomb
blockade signatures, but dramatically different topological orders and braiding
statistics.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics