  Part of the CMB polarization signal in the direction of galaxy clusters is
produced by Thomson scattering of the CMB temperature quadrupole. In principle
this allows measurement of the CMB power spectrum harmonic $C_2(z)$ with higher
accuracy (at $z>0$) than the cosmic variance limit imposed by sample variance
on one CMB sky. However the observed signals are statistically correlated if
the comoving separation between the clusters is small enough. Thus one cannot
reduce the sample variance by more than roughly the number of separate regions
available which produce uncorrelated signals, as first pointed out by
Kamionkowski and Loeb. In this paper we analyze in detail the procedure
outlined by Kamionkowski and Loeb, computing the correlation of the
polarization signals by considering the variation of the spherical harmonic
expansion coefficients of the temperature anisotropy on our past light cone. We
find that in fact the cosmic variance is not reducible below the single sky CMB
value using the cluster method. Thus this method is not likely to be of use for
reconstruction of the primordial power spectrum. However the method does yield
a measurement of $C_2$ as a function of redshift with increasing accuracy at
higher redshift, and thus potentially a probe of the mechanism which may have
suppressed the quadrupole. (abridged)
