The sub-conscious mind is to consciousness what the quantum world is to the classical world.
The vast majority of brain activity is non-conscious; consciousness is "the tip of an iceberg" of neural activity. Yet the threshold for transition from pre-, non-, or sub-conscious processes into conscious awareness is unknown. The sub-conscious mind as revealed in dreams has been described by Matte Blanco as a place where "paradox reigns, and opposites merge to sameness". Reality is seemingly described by two separate sets of laws. In our everyday classical world, Newton's laws and Maxwell's equations accurately portray reality. However at small scales, the bizarre laws of quantum mechanics rule: particles are distorted in space and time (uncertainty), exist in multiple states or locations simultaneously (superposition) and remain connected in opposite states over distance (nonlocal entanglement). In the quantum world "paradox reigns and opposites merge to sameness".
The boundary, or threshold between the quantum and classical worlds (i.e. quantum state reduction, collapse of the wave function, measurement, decoherence) remains mysterious. Early quantum theorists attributed reduction/collapse to observation: "consciousness collapses the wave function". Modern physics attributes reduction/collapse to any interaction with the classical environment ("decoherence"). Neither solves the problem of isolated quantum superpositions which are nonetheless useful in quantum computation.
In quantum computation, information may be represented as isolated superpositions (e.g. as quantum bits—"qubits"—of both 1 AND 0) which interact/compute by nonlocal entanglement, and eventually reduce/collapse to classical solutions.
Based on a 1989 suggestion by Sir Roger Penrose, he and I have put forth a specific model of consciousness involving quantum computation in microtubules within the brain's neurons. Superpositions of multiple possible pre-/sub-conscious perceptions or choices reach threshold for self-collapse (by Roger's "objective reduction" due to properties of fundamental spacetime geometry), and select/reduce to particular classical perceptions or choices. Each reduction is a conscious event, a series of which gives a "stream of consciousness".
The main scientific objection to our proposal has been that the brain is too warm for quantum computation which in the technological realm seems to require ultra cold temperatures to avoid thermal decoherence. However recent evidence shows that quantum processes in biological molecules are enhanced by increased temperature. Evolution has had billion of years to solve the problem of decoherence. Consciousness may be a particular form of quantum state reduction: a process on the edge between the quantum and classical worlds.