Just mentioning some points here.
Ancient Greece, as is commonly perceived (foundation of language, philosophy, poetry and arts, culture and all modern sciences) ended as such in 335BC, when Thebes were destroyed to the ground by Alexander the Macedon (Alexander the Great).
Historically, Ancient Greece was never an empire similar to Roman Empire, Persian Empire, British Empire etc. It is some times considered as an "Empire" falsely, maybe because the influence of Ancient Greece then and now is greater than any other civilization.
Alexander the Great wasn't even Greek, he was Macedonian. He never "spread" the Ancient Greek civilization to the East. Greek was the prominent language already, as now is the English language and was spoken in all of the known world due mainly to business interactions between different civilizations. He himself not being Greek had Aristotle -one of the major figures in (Greek) philosophy - as a teacher until his early teens.
The Hellenistic period already differed from Ancient Greece, mainly because the language changed severely and has little in common with Ancient Greece, apart from developing in the more or less same geographic area.
Byzantine Empire (Roman Empire of the East) was far from Greek (culturally or otherwise). Religion was totally different (Ancient Gods/ Christianity), Greek was just the common/ official language until then (much different from the Ancient Greek language), the majority of Byzantine Emperors (Roman Emperors of the East) were Thracians (not Greeks) and Illyrians (not Greeks, today's Albania).
I think the OP question is false. Maybe Roman Empire should be compared with the Persian Empire, the Chinese Empire, the Indian Empire and such all of which were far superior than the Roman Empire in my opinion.
Goin' back to poker now