  The ``low'' (hard or ``non-thermal'') state of black hole candidates is
sometimes modelled via an optically thick, hot Compton cloud that obscures a
softer input source such as an accretion disk. In these models the observed
output spectra consist entirely of photons reprocessed by the cloud, making it
difficult to extract information about the input spectra. Recently Miller
(1995) has argued that the Fourier phase (or time) lag between hard and soft
X-ray photons in actuality represents the phase lags intrinsic to the input
source, modulo a multiplicative factor. The phase lags thus would be a probe of
the input photon source. In this paper we examine this claim and find that,
although true for the limited parameter space considered by Miller, the
intrinsic phase lag disappears whenever the output photon energy is much
greater than the input photon energy. The remaining time lags represent a
``shelf'' due to differences between mean diffusion times across the cloud. As
pointed out by Miller, the amplitude of this shelf -- which is present even
when the intrinsic time lags remain -- is indicative of the size and
temperature of the Compton cloud and is a function of the two energies being
compared. However, we find that with previous instruments such as Ginga the
shelf, if present, was likely obscured by counting noise. A more sensitive
measure of Compton cloud parameters may be obtainable from the coherence
function, which is derived from the amplitude of the Fourier cross power
spectral density. This function has been seen to exponentially decrease at high
Fourier frequencies in Cygnus X-1. Coherence loss is characteristic of Compton
clouds that undergo large variations of size and/or temperature on time scales
longer than about 10 seconds. We argue that observing phase lags and coherence
