  Following the previous works on the A. Pr\'astaro's formulation of algebraic
topology of quantum (super) PDE's, it is proved that a canonical Heyting
algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE.
This is directly related to the structure of its global solutions. This allows
us to recognize a new inside in the concept of quantum logic for microworlds.
Furthermore, the Prastaro's geometric theory of quantum PDE's is applied to the
new category of {\em quantum hypercomplex manifolds}, related to the well-known
Cayley-Dickson construction for algebras. Theorems of existence for local and
global solutions are obtained for (singular) PDE's in this new category of
noncommutative manifolds. Finally the extension of the concept of exotic PDE's,
recently introduced by A.Pr\'astaro, has been extended to quantum PDE's. Then a
smooth quantum version of the quantum (generalized) Poincar\'e conjecture is
given too. These results extend ones for quantum (generalized) Poincar\'e
conjecture, previously given by A. Pr\'astaro.
