re: Everyone agrees the ancient Romans were the greatest ancient civilization....

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You forgot to include military tactics and organization.

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Not really. The Roman legion was developed from the existing greek phalanx. What Rome did was break the phalanx from a total infantry formation into effectively "small scale" phalanxes that were able to be moved independently of each other. They gained flexibility.

When the phalanx and the legion actually fought, the greeks didn't use Alexander's tactics (of using cavalry to control the flanks), instead just throwing their phalanx at the roman legions and cavalry... so the romans kicked their asses. If the greeks had been able to use cavalry to control the flanks, the phalanx would have destroyed the legion.

Probably the biggest advance from helenistic combat to roman combat was that helenistic commanders were at the front fighting, while roman commanders gave orders from the rear. This allowed them to better use the flexibility of the legions over the strength of the phalanx head on by being able to regularly outflank the greeks once Alexanders tactics of cavalry flanking had been abandoned.

The legion gained advantage any time they could force the greeks to abandon their long spears and fight with their swords (kopis). While the massed long spears could penetrate the roman shields, those shields were far superior to greek shields that gave the romans a huge advantage in hand to hand.

Regardless, neither would have stood up well to the Han armies in china. They had crossbows that would penetrate any armor that was available at the time. The crossbow didn't make it's way to europe until the 12 century. In the second century, the chinese army already had 50,000 crossbowmen.

We tend to be very western focused in what we learn about... but for a long time China was well ahead of the west in terms of technology, including military technology. China had cannon two centuries before it made it's way to europe.