When considering the possible relationship between quantum theory and spirituality, one first needs to agree on basic definitions. Quantum theory, elaborate and strange as it is, can be readily defined through its principles and equations. Spirituality is another matter altogether.
Dictionaries and online sources, like Wiktionary, define the term variously as concern for that which is unseen, intangible, or noncorporeal. If we were to say that spirituality is concern for matters of the soul, we would be begging the question, since "soul" and "spirit" are close synonyms and difficult to pin down in their own right. If we accept dictionary definitions that aren't circular, we should be on safer ground.
The key word in the definition, it seems to me, is "concern," a state of consciousness where one's attention is drawn to things that are nonphysical. What the definitions leave out is that the result of this attention often has profound effects on one's feelings, emotions, and ideas, whether it's a belief in a single God or many gods, various forms of transcendentalism, or any state of consciousness that is experienced as extraordinary and personally compelling.
It is natural that some (or many) would argue for the wholesale separation of spirituality from science - in this case, quantum theory. In the West, we've been trained for several hundred years in all manners of reductionism, taking the physical world apart and reducing its components to the smallest size possible and putting everything into separate boxes. Matters of the soul and matters of the atom should follow this tradition and share no common ground. One is diffuse, indescribable, unquantifiable. The other is...(oops!) also diffuse, indescribable, unquantifiable (at least in part).
The arrival of quantum theory in the early 20th Century effectively killed reductionism in physics once and for all. The tiniest inseparable things then known - protons, neutrons, electrons and photons - could not be seen, put in a box, or even fully measured. A single photon itself had a split personality: it was a "wave" of energy with a mathematical function until it was observed or measured, when it transformed itself into a single particle. A single photon also had the magical power to move along any number of possible wayward routes to get to its destination - a wave of probability - until it finally hit a detecting device and was observed.
The most widely embraced view of the implications of quantum theory, the Copenhagen Interpretation, makes the observer an integral part of the system. With no observation, with no measurement, particles move around in a cloud of varying degrees of probability, a quantum state known as coherence. They "decohere" only when a conscious being enters the fray and decides to take a measurement. Physics Nobelist Eugene Wigner wrote of this interpretation, " It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness."
It is now widely accepted that consciousness itself cannot be explained by the old physics of "classical mechanics." Several leading physicist and mathematicians now subscribe to Quantum Mind Theory (or quantum consciousness theory) that sees some features of quantum mechanics operating inside the brain. These include entanglement, where two paired particles instantaneously respond to each other's changing states, and superposition, where a particle can be in two different states or locations at once. Similarly, mathematician Roger Penrose has argued, in his books The Emperor's New Mind" and "The Shadows of the Mind," that there are things going on in the brain that can't be demonstrated or characterized by any mathematical operation; in other words, no computer, no matter how enormous and powerful, can compete with what goes on inside the human mind. He has determined that quantum effects are at work inside all the microtubules of our brains.
Others, like physicist David Bohm and botanist Rupert Sheldrake, spent much of their lives working out different field theories with powerful implications for human consciousness. Bohm posited that the everyday world around us is the "explicate order," and that everything else (the unseen, the intangible, the noncorporeal - the stuff of spirituality) are part of an "implicate order" of a limitless ocean of simultaneous interconnections. Sheldrake, for his part, was mystified by some of his work in plant morphology - how plants "know" to grow and form themselves a certain way. Individual plant shape, he determined, could not be fully orchestrated by its DNA. He argued instead that plants, in fact all living creatures, connect in some way with a "morphogenetic field" of information - a field that is inaccessible to our abilities to perceive it. "Morphic resonance" flows back and forth between the organism and the invisible field, each acquiring information from the other. Organisms grow and change a certain way, in part, because of this transfer of information.
As a special and heightened form of consciousness, spirituality cannot be isolated from science. In fact, the potential for mutual entanglement is so rich that it would seem, at best, parochial for humans to assume that their spirituality is a purely standalone phenomenon completely separate from what's happening at the very smallest scale of the physical world.