  Recent experiments suggest a quantum spin liquid ground state in the material
PbCuTe$_2$O$_6$, where $S=1/2$ moments are coupled by antiferromagnetic
Heisenberg interactions into a three dimensional structure of corner sharing
triangles dubbed the hyper-hyperkagome lattice. It exhibits a richer
connectivity, and thus likely a stronger geometric frustration, than the
relatively well studied hyperkagome lattice. Here, we investigate the possible
quantum spin liquids in the $S=1/2$ hyper-hyperkagome magnet using the complex
fermion mean field theory. Extending the results of a previous projective
symmetry group analysis, we identify only two $\mathbb{Z}_2$ spin liquids and a
$U(1)$ spin liquid that are compatible with the hyper-hyperkagome structure.
The $U(1)$ spin liquid has a spinon Fermi surface. For the $\mathbb{Z}_2$ spin
liquids, one has a small excitation gap, while the other is gapless and
proximate to the $U(1)$ spin liquid. We show that the gapped and gapless spin
liquids can in principle be distinguished by heat capacity measurements.
Moreover, we calculate the dynamical spin structure factors of all three spin
liquids and find that they highly resemble the inelastic neutron scattering
spectra of PbCuTe$_2$O$_6$. Implications of our work to the experiments, as
well as its relations to the existing theoretical studies, are discussed.
