The first thing to say is that all such precognitive theories are, with at least some justification, regarded by the mainstream as pseudoscience. While we have learned that the trio of temporal causality, locality and realism cannot all be aspects of the underlying reality and we have no idea which is or are not, there are a good many theorems in quantum theory as to why the known forms of "quantum weirdness" cannot transport information backwards in time.
Having said that, several theories along the lines you propose have been put forward. First off the block, in 1927, was JW Dunne in An Experiment with Time (many subsequent editions). He proposed that our consciousness extends forward only along its own timeline, i.e. we can foresee or, in his case, dream of, only our own experiences in the future. We mostly don't see it because the brain is so busy paying attention to the present moment. He decided that precognition of near-future events was far more common and that quantum uncertainties provided the clue as to how it all faded off in the far future, but QM was not yet developed enough to say anything sensible. He also proposed infinite regresses in levels of time and consciousness. Dunne's theory was discussed at the time by philosophers of Time such as JA Gunn, MF Cleugh and CD Broad, but was not widely taken up and his infinite regresses were universally condemned.
A more recent proponent, describing almost exactly your model, is psychologist Fernando de Pablos. See for example his Brain at Night: Scientific Foundations of Precognitive Dreams, Diego Marin, 2011, pages 236-7 (If you can get hold of a copy). However I find his arguments to be disorganized and unscientific. He also appears to have little expertise in QM. He cites several earlier books and papers of his (which I have not seen). Two which look like they might be relevant here are:
- "Spontaneous Precognition During Dreams: A Theoretical Model", ''J. Society for Psychical Research, Vol. 68, No. 4, 2004. pp.226-244.
- Precognitive Memory, Diego Marin, 2006.
Having said all that, quantum theory remains incomplete and continues to get ever weirder, while information theory is transcending its roots in telecommunications theory and beginning to influence the philosophy of artificial and even human intelligence. It is conceivable that the likes of Penrose may be on to something (though I regard his microtubule theories as lacking in a coherent philosophy of mind), and that some as-yet undiscovered quantum weirdness might allow conscious information to leak back in time.
But none of the theorists mentioned is or was a philosopher of any standing. I know of none such who have run with this kind of model.