OK, thanks.
So, we have at least one evidence (from the originator of this topic) that changing ONT + splitter + connection to ONT helped to solve the same problem.
I believe I have no choice but request a verizon technician to stop by and either fix the problem (by changing ONT, or whatever else) or, take this Quantum stuff back, so I'll be having fun time exploring alternatives to FiOS...
Why it could not just work? : - )