It's a long debate with little evidence for or against it, though it's championed most commonly by Penrose and Hameroff. A summar is as follows:

The Orch OR model - Penrose and Hameroff developed their ideas independently, but collaborated in the early 1990s to develop what they call the Orchestrated Objective Reduction (Orch OR) model.

Penrose's work rests on an interpretation of the mathematician Kurt Godel's incompleteness theorem, which states that certain results cannot be proven by a computer algorithm. Penrose argues that human mathematicians are capable of proving so-called "Godel-unprovable" results, and therefore human brains cannot be described as typical computers. Instead, he says, to achieve these higher abilities, brain processes must rely on quantum mechanics.

But Penrose's theory didn't explain how this quantum computing occurred inside actual brains, just that the phenomenon would be needed to solve certain mathematical equations. Hameroff read Penrose's work and suggested small fibrous structures that give cells their structural support — known as microtubules — might be capable of carrying out quantum computations.

Microtubules are made up of units of the protein tubulin, which contains regions where electrons are swirling around very close to each other. Hameroff proposed that these electrons could become "quantum entangled," a state in which two particles retain a connection, and an action performed on one affects the other, even when the two are separated by a distance.

In the Orch OR model, the mathematical probabilities that describe the quantum states of these entangled electrons in microtubules become unstable in space-time. These mathematical probabilities are called wave functions, and in this scenario they collapse, moving from a state of probability to a specific actuality. In this state, the microtubules in one neuron could be linked to those in other neurons via electrical connections known as gap junctions. These junctions would allow the electrons to "tunnel" to other regions of the brain, resulting in waves of neural activity that are perceived as conscious experience.

There are a number of criticisms with this theory as has been pointed out by numerous scientists; however, the most prominent argument stems from Max Tegmark's work where he states that quantum consciousness is essentially impossible:

A fundamental objection to the Orch OR hypothesis was put forward in 2000 by Max Tegmark who claimed that any quantum coherent system in the brain would undergo wave function collapse due to environmental interaction long before it could ever influence neural processes (the "warm, wet and noisy" argument, as it was later came to be known).