While substantial progress has been made in the field known as artificial consciousness, at the present time there is no generally accepted phenomenally conscious machine, nor even a clear route to how one might be produced should we decide to try. Here, we take the position that, from our computer science perspective, a major reason for this is a computational explanatory gap: our inability to understand/explain the implementation of high-level cognitive algorithms in terms of neurocomputational processing. We explain how addressing the computational explanatory gap can identify computational correlates of consciousness. We suggest that bridging this gap is not only critical to further progress in the area of machine consciousness, but would also inform the search for neurobiological correlates of consciousness and would, with high probability, contribute to demystifying the “hard problem” of understanding the mind–brain relationship. We compile a listing of previously proposed computational correlates of consciousness and, based on the results of recent computational modeling, suggest that the gating mechanisms associated with top-down cognitive control of working memory should be added to this list. We conclude that developing neurocognitive architectures that contribute to bridging the computational explanatory gap provides a credible and achievable roadmap to understanding the ultimate prospects for a conscious machine, and to a better understanding of the mind–brain problem in general.