Roger Penrose and Stuart Hameroff have argued that consciousness is not a consequence of interactions between neurons in the brain but arises as from smaller structures such as microtubules within cells, for which quantum effects could be significant.
Microtubules are cylindrical structures that self-assemble to form the shape of the cells that form the brain. They're also the nervous system of the cell and process information internally to organize what happens within each cell and also how cells interact with other cells. These microtubules are actually very well designed as computational devices. They are tiny on-board computers which organize cellular activities within the brain. They act as quantum computers.
The quantum world, at the level of atoms and below has some very strange properties. For example, everything can be interconnected to everything else, particles can be in two or more places at the same time (a process called "superposition"). So in a quantum computer, information can be in two states at the same time. This ability to be in two places at once (known as "superposition" and "space-time geometry") is thought to be a fundamental property and the very fabric of the universe.
The fundamental level of the universe is so infinitesimally small that it's impossible to even imagine. But if we go down inside our brains to our nerve cells and into the microtubules and inside the microtubule subunits to the level of atoms and then keep going down even smaller than atoms (which are mostly empty space), the space between the nucleus and the electrons, down and down and down, everything is smooth. But eventually we reach a level where there's some kind of coarseness or irregularity. This may be something like being in an airplane looking at the surface of the ocean from 33,000 feet. The surface of the ocean looks very smooth. However, if you were on a boat on that surface, it'd be choppy and there's a pattern in the waves in the surface of the ocean. Similarly when we get down to the fundamental level of the universe, there's information. There's patterns. And that information carries conscious precursors that give rise to our complex conscious.
Simply put, consciousness is an emergent property of computation inside the brain's neurons to the level of microtubules. It's a form of quantum computation and a manifestation of the quantum cytoskeletal state and its interplay between quantum and classical levels of activity. Protoconscious information is encoded in space-time geometry at the fundamental Planck scale and a self-organizing Planck-scale process results in consciousness.
In other words, it is at the most basic level of the universe, deep inside the brain that consciousness occurs. As Einstein's General Theory of Relativity told us...everything - matter, energy, space, and time are particular arrangements. They are the fabric of the universe, which works on the tiniest of scales. It's at this level that consciousness exist, connected to the brain by quantum processes in the microtubules inside nerve cells.
If you search for the terms <"quantum computer" microtubule> in the biomedical literature, you will find the article, "Importance of quantum decoherence in brain processes" by Max Tegmark. In that article, Tegmark concludes that it is not likely that the brain acts as a quantum computer and, "cognitive processes should be thought of as a classical rather than quantum system". If the fans of "quantum consciousness" could demonstrate that brain microtubules "act as quantum computers" then the "quantum consciousness" field might move beyond the protoscience stage.
Microtubule article at Wikipedia