Why doesn't H₄CO₄ exist as a stable compound?
Wikipedia reports that calculations show orthocarbonic acid spontaneously decomposes to the ordinary (meta)carbonic acid $\ce{H2CO3}$ plus water.  However, orthocarbonate esters $\ce{C(OR)4}$ do exist.

The orthoacid may, however, become stable at high pressure [1, with a nontechnical summary given in 2] and possibly exist within Uranus and Neptune.  It is further posited that $\ce{H2CO3}$, ordinarily stable only in aqueous solution, becomes a stable solid in the interiors of icy moons such as Europa, possibly impacting interactions thought to be necessary for the development of subterranean life.

A final note: the swastika-like shape of the orthocarbonic acid molecule has led unfortunately to the name "Hitler's acid".  Actually the swastika long predates the Nazis and retains non-Nazi usages to this day.