As one example; taken as a whole, the work of parapsychologists over several decades in the mid-late 20th century to measure psi ability by laboratory experiment has been a failure. Those experiments that have allegedly shown statistically significant results to prove (say) ESP ability have not been able to be repeated and have therefore been written off. Also, Dr Susan Blackmore and others have exposed shonky experimental procedure and even outright fudging of results in some cases. I'm fine with that because I think it unlikely that ESP could be shown to occur under such mundane, repetitive testing conditions. But let's say that ESP does happen in more emotionally extreme and less predictable scenarios, such as with 'crisis apparitions' - a commonly reported and well documented phenomenon and one that ESP could explain.
The scientists agree there is no such thing as ESP. Yet they cannot understand what human consciousness is. Consciousness is a crucial aspect of Quantum theory. Quantum is the strongest theory in science, yet it requires a 'conscious observer' to work. Physics has (historically) tried to escape this reality by means such as the Copenhagen Interpretation of Quantum Mechanics (way too complex to describe here, and the Wikipedia article on it is quite heavy going and extremely counter-intuitive, as is everything to do with QM), but this is a cop out. Another cop out is to say that the mind is nothing more than a creation of the workings of the brain. That is the Materialist argument, one that has been built upon the ability of classical (Newtonian) physics to describe the physical operations of the brain. The human mind is a way too complex, individually varied and changing thing to yield its secrets to such a simplistic and Deterministic viewpoint. Can physics explain how and why we make and appreciate art?
What if, say, the mind is the observer and the brain is a physical system in a state of more-or-less constant quantum superposition? What if the mind can Influence (Niels Bohr's word) the brain (or other minds) using what Einstein uncomfortably-termed 'spooky interactions at a distance'? (See Quantum Entanglement.) These are just a few examples of the sort of questions that QM throws up. If ESP does actually occur, a Quantum understanding of the brain and consciousness would likely allow room for it to do so and explain how it is possible.
This would change everything. And once one skittle falls, more will follow.
Other than via the links provided, if you want to read a well-written and thoughtful book, I'd recommend 'Quantum Enigma - Physics Encounters Consciousness', by Bruce Rosenblum & Fred Kuttner, as a good starting point (1) for reading and research into this fascinating field.
By James Gilberd, co-author of Spooked - Exploring the Paranormal in New Zealand, and leader of the Strange Occurrences group.
(1) Rosenblum & Kuttner are skeptics of the paranormal and frown upon dilatantes like me fumbling about in the field of QM in search of explanations for paranormal phenomena! (See their 'Paraphenomena' on page 192 of 'Quantum Enigma'.)