Is spacetime an illusion? In consistent histories, for gauge theories, can the projection operators used in the chains be not gauge invariant?
In quantum gravity, for a projection operator to be gauge invariant means it has to be diffeomorphism invariant. There are no localized diffeomorphism invariant operators, not even localized in time. No localized observables.
So, what are the permissible chains in quantum gravity? Do they have to be delocalized, or localized on the boundaries of spacetime?
Are space and time illusions, as Andrew Strominger claimed? A Cosmic Hologram projecting out the illusion that is spacetime from the future boundary of time?
 A: Gauge invariance does not imply diffeomorphism invariance unless one confounds the fictitious spacetime metric with the physical metric (this kind of confusion is very common in the literature) [1].
As a consequence, the generator of time translations is not zero and time is perfectly valid in a quantum gravity context (no problem of absence of time).
Space is also valid and has associated (hermitian) operators in quantum gravity. However this does not mean that spacetime is fundamental, because the $x^0$ in spacetime is not the causal concept of time (Strominger confounds both concepts).
The correct term is "emergent" not "illusion". Precisely I wrote something about the emergence of spacetime in FQXI:
Eight Assumptions of Modern Physics Which Are Not Fundamental
See also Pavsič's book for a distinction between universe evolution time (denoted by $\tau$) and $x^0$:
The Landscape of Theoretical Physics: A Global View; 2001; Kluwer, Dordrecht; Pavsič, M. 

[1] The former is built from the flat metric plus the potential of the gauge field. The latter is built from a fictitious background metric plus a deviation, and this deviation differs from the gauge potential due to higher-order graviton corrections.
