  We provide several new results on quantum state space, on lattice of
subspaces of an infinite dimensional Hilbert space, and on infinite dimensional
Hilbert space equations as well as on connections between them. In particular
we obtain an n-variable generalized orthoarguesian equation which holds in any
infinite dimensional Hilbert space. Then we strengthen Godowski's result by
showing that in an ortholattice on which strong states are defined Godowski's
equations as well as the orthomodularity hold. We also prove that all 6- and
4-variable orthoarguesian equations presented in the literature can be reduced
to new 4- and 3-variable ones, respectively and that Mayet's examples follow
from Godowski's equations. To make a breakthrough in testing these massive
equations we designed several novel algorithms for generating Greechie diagrams
with an arbitrary number of blocks and atoms (currently testing with up to 50)
and for automated checking of equations on them. A way of obtaining complex
infinite dimensional Hilbert space from the Hilbert lattice equipped with
several additional conditions and without invoking the notion of state is
presented. Possible repercussions of the results to quantum computing problems
are discussed.
