Speaking for my own area, generally speaking they didn't. So far as we can tell, anyway. The ancient Greeks had a well-defined sense of the past, and of "how we got here", even if that sense was based on largely mythical stories rather than on anything accurate. But there were only a couple of narratives floating around that could lend themselves to being extrapolated into the future.
The more famous one is the "Myth of the Races". This is a cross-cultural myth whose origins we associate with the Near Eastern Bronze Age, in which several metals or other materials represent different kings or periods. In the Greek variants it takes on a cosmic timespan, with each metal representing an age of unspecified length, and with the "present" being represented by iron or bronze. The most famous version is the one in the Hesiodic Works and Days (ca. 700-660 BCE), but the real thrust of the story is best expressed in a poem from several hundred years later, Aratos' Phainomena lines 100-136: his version is the one that most explicitly ties the different metals to a progressive deterioration in the ethics and character of humanity (English translation here).
The Hesiodic version doesn't have a straightforward narrative of degradation like Aratos does, but the same pessimism does come out clearly in the last of the Hesiodic races, the Iron Race. Here's how the poet imagines the future (W&D 180-201, tr. Most):
But Zeus will destroy this race of speech-endowed human beings too, when at their birth the hair on their temples will be quite gray. Father will not be like-minded with sons, nor sons at all, nor guest with host, nor comrade with comrade, nor will the brother be dear, as he once was. They will dishonor their aging parents at once; they will reproach them, addressing them with grievous words-cruel men, who do not know of the gods' retribution!-nor would they repay their aged parents for their rearing. Their hands will be their justice, and one man will destroy the other's city. Nor will there be any grace for the man who keeps his oath, nor for the just man or the good one, but they will give more honor to the doer of evil and the outrage man. Justice will be in their hands, and reverence will not exist, but the bad man will harm the superior one, speaking with crooked discourses, and he will swear an oath upon them. And Envy, evil-sounding, gloating, loathsome-faced, will accompany all wretched human beings. Then indeed will Reverence and Indignation cover their beautiful skin with white mantles, leave human beings behind and go from the broad-pathed earth to the race of the immortals, to Olympus. Baleful pains will be left for mortal human beings, and there will be no safeguard against evil.
Alongside this narrative of human development we also have a narrative of technological progress. Here's a snippet from Sophokles' play Antigone, lines 353-75:
[Man] has learned language, thoughts
as swift as wind
and social feelings; he has learned how to escape
from sharp arrows of bitter frost
under clear skies, and darts of driving rain.
He’s always ready -- and he never tries to tackle anything
unless he is. Only from Death
will he not find release;
he has devised a cure even
for desperate diseases.
We get similar sentiments in many other places. Anaxagoras fr. B 21 talks about the development of animal husbandry as a purposed idea; Demokritos fr. B 144 talks about music being a recent development that came about because of superfluity of resources. Some poets attribute technology to divine inspiration rather than human resourcefulness, but the gods often served as a kind of metaphor for physical processes, so it's not necessarily something totally different: the Homeric Hymn to Hephaistos (prob. late 5th century BCE) talks about the "glorious works" that humans have learned from Hephaistos. In the play Prometheus Bound, lines 436-506 (5th century), Prometheus talks about how humans were without sense or reason before he came along, but then he taught humanity architecture, astronomy, mathematics, writing, animal husbandry, medicine, reading of dreams and omens, religious liturgy, and metallurgy. And the same idea comes up again in Euripides' Suppliants, lines 201-213.
In principle this technological narrative could be extrapolated into the future. But we don't have any sign that any ancient Greeks ever did. We don't get people writing about future technological developments. Like I said at the start of this post, it's a narrative of "how we got here", not one of an ongoing process.