The fundamentals of Lukasiewicz-Moisil logic algebras and their applications to complex genetic network dynamics and highly complex systems are presented in the context of a categorical ontology theory of levels, Medical Bioinformatics and self-organizing, highly complex systems. Quantum Automata were defined in refs.[2] and [3] as generalized, probabilistic automata with quantum state spaces [1]. Their next-state functions operate through transitions between quantum states defined by the quantum equations of motions in the SchrÄodinger representation, with both initial and boundary conditions in space-time. A new theorem is proven which states that the category of quantum automata and automata-homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)-Systems which are open, dynamic biosystem networks [4] with de¯ned biological relations that represent physiological functions of primordial(s), single cells and the simpler organisms. A new category of quantum computers is also defined in terms of reversible quantum automata with quantum state spaces represented by topological groupoids that admit a local characterization through unique, quantum Lie algebroids. On the other hand, the category of n-Lukasiewicz algebras has a subcategory of centered n-Lukasiewicz algebras (as proven in ref. [2]) which can be employed to design and construct subcategories of quantum automata based on n-Lukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref. [2] the category of centered n-Lukasiewicz algebras and the category of Boolean algebras are naturally equivalent. A `no-go' conjecture is also proposed which states that Generalized (M,R)-Systems complexity prevents their complete computability (as shown in refs. [5]-[6]) by either standard, or quantum, automata.