The computational significance of consciousness is an important and potentially more tractable research theme than the hard problem of consciousness, as one could look at the correlation of consciousness and computational capacities through, e.g., algorithmic or complexity analyses. In the literature, consciousness is defined as what it is like to be an agent (i.e., a human or a bat), with phenomenal properties, such as qualia, intentionality, and self-awareness. The absence of these properties would be termed “unconscious.” The recent success of large language models (LLMs), such as ChatGPT, has raised new questions about the computational significance of human conscious processing. Although instances from biological systems would typically suggest a robust correlation between intelligence and consciousness, certain states of consciousness seem to exist without manifest existence of intelligence. On the other hand, AI systems seem to exhibit intelligence without consciousness. These instances seem to suggest possible dissociations between consciousness and intelligence in natural and artificial systems. Here, I review some salient ideas about the computational significance of human conscious processes and identify several cognitive domains potentially unique to consciousness, such as flexible attention modulation, robust handling of new contexts, choice and decision making, cognition reflecting a wide spectrum of sensory information in an integrated manner, and finally embodied cognition, which might involve unconscious processes as well. Compared to such cognitive tasks, characterized by flexible and ad hoc judgments and choices, adequately acquired knowledge and skills are typically processed unconsciously in humans, consistent with the view that computation exhibited by LLMs, which are pretrained on a large dataset, could in principle be processed without consciousness, although conversations in humans are typically done consciously, with awareness of auditory qualia as well as the semantics of what are being said. I discuss the theoretically and practically important issue of separating computations, which need to be conducted consciously from those which could be done unconsciously, in areas, such as perception, language, and driving. I propose conscious supremacy as a concept analogous to quantum supremacy, which would help identify computations possibly unique to consciousness in biologically practical time and resource limits. I explore possible mechanisms supporting the hypothetical conscious supremacy. Finally, I discuss the relevance of issues covered here for AI alignment, where computations of AI and humans need to be aligned.