  Recent quantum oscillation experiments in underdoped high temperature
superconductors seem to imply two paradoxes. The first paradox concerns the
apparent non-existence of the signature of the electron pockets in angle
resolved photoemission spectroscopy (ARPES). The second paradox is a clear
signature of a small electron pocket in quantum oscillation experiments, but no
evidence as yet of the corresponding hole pockets of approximately double the
frequency of the electron pocket. This hole pockets should be present if the
Fermi surface reconstruction is due to a commensurate density wave, assuming
that Luttinger sum rule relating the area of the pockets and the total number
of charge carriers holds. Here we provide possible resolutions of these
apparent paradoxes from the commensurate $d$-density wave theory. To address
the first paradox we have computed the ARPES spectral function subject to
correlated disorder, natural to a class of experiments relevant to the
materials studied in quantum oscillations. The intensity of the spectral
function is significantly reduced for the electron pockets for an intermediate
range of disorder correlation length, and typically less than half the hole
pocket is visible, mimicking Fermi arcs. Next we show from an exact transfer
matrix calculation of the Shubnikov-de Haas oscillation that the usual disorder
affects the electron pocket more significantly than the hole pocket. However,
when, in addition, the scattering from vortices in the mixed state is included,
it wipes out the frequency corresponding to the hole pocket. Thus, if we are
correct, it will be necessary to do measurements at higher magnetic fields and
even higher quality samples to recover the hole pocket frequency.
