So we're not looking at these states on their own merits but on the merits of what subsequent generations did with their (supposed*) cultural legacies...?
We should be looking at them through several angles. While it is of course good to look at these states through a military and political angles the culture of the state should be observed as well. Even taking these away the military power of the Byzantines, the Commercial prowess and administrative brilliance of the Arabs, and the political power the Franks held in Europe still hold large arguments for their inclusion. I understand the complaining about Eurocentrism, and it has merit, but taking these away altogether is absurd.
I'm not sure there were immense gains in science or literature. And even were there I'm not sure what immense gains would be like, seem like or show up in the historical record like. So I suppose the claim should be put in relation to some other period perhaps.
Algebra, scientific method, Trigonometry, the transmission of classical learning to the rest of Europe through Spain(As evidenced by the use of Spanish translations of the Bible from Greek in Italy), large advances in Astronomical thinking. I could go on, of course. While most of Europe was in a technological stasis during the early middle ages the Arabs were making great strides in every direction. There was even a paper factory in Baghdad after the Battle of Talas.
I've said it about a dozen times in a half dozen languages: Srivijaya. I'll make the case that it transmitted Buddhist works from India to China and was a major site of Buddhist learning, so much so, that Indian monks seem to have made a point to go to Srivijaya to copy down Buddhist tracts that weren't available on the mainland. It was also fabulously rich and had a profound religious and cultural influence on the whole of the Malay archipelago. While I'm at it I could put down the Khmer for their religious and cultural influence on well themselves, the Tai and Burmans. Much as I dislike the Khmer for stealing ma research dollars, I can at least respect them for being hugely influential across the region.
Srivijaya is a good choice and I would lobby for it to be included in the poll. It was a very powerful polity in Southeast Asia and its cultural influence as well as commercial power cannot be denied.
Axum by this time, had unfortunately, declined, for several reasons. Not only did the Arab conquests isolate it from Indian sea trade, but climate change also caused a loss of fertility and caused people to move further inland. In Africa, Ghana, as Huayna said, is a good choice.