This, he argues, is a "structural reality" that has to be faced by the US, and the world.
Further, with China's huge market and economy, comes what Prof Allison calls "geo-economic power", the ability to hold sway over other countries seeking to participate in its surging economic growth.
But for generations of Americans brought up on the idea of the US being at the top of the pecking order on many fronts, with all the power and perquisites that this entails, the new reality comes as a shock. It leaves many discomforted about what it all portends for them, and the world.
"The US should stop playing, 'let's pretend'", he says pointedly in his book. "Instead, it needs to take the economic and strategic challenge from China seriously, start investing in boosting its economy and developing its technological capabilities, because the status quo cannot be sustained when the underlying economic balance of power has tilted so dramatically in China's favour."
With the same candour, the Harvard professor points to the geopolitical implications that arise from China's relentless economic march. Putting it starkly, he notes that, for some in Beijing, "Making China Great Again" entails:
- Returning China to the predominance in Asia that it enjoyed before the West intruded.
- Re-establishing control over the territories of "greater China", including not just Xinjiang and Tibet on the mainland, but also Hong Kong and Taiwan.
- Recovering its historic sphere of influence along its border and in the adjacent seas so that others give it the deference great nations have always demanded.
- Commanding the respect of other great powers in the councils of the world.
He adds in his book: "At the core of these national goals is the civilisation creed that sees China as the centre of the universe... In this narrative, the rise of the West in recent centuries is a historical anomaly, reflecting China's technological and military weakness when it faced dominant imperial powers. Xi Jinping has promised his fellow citizens: no more."
End of the illusion
The view of China as a rising and revisionist power is now widely held in Washington circles, says Prof Allison. Indeed, I found this to be so in many conversations I had with business, political and academic leaders in the US while on a recent visit.
The previous consensus, advanced by the Obama administration, that drawing China into the international system would make it a responsible stakeholder - or strategic partner - in the global order is now regarded as overly optimistic, if not downright naive. It has given way to a new view of China as a strategic rival to the US, out to displace it from its present perch.
This deep shift in thinking straddles the political divide. It pre-dates, and will outlast, Mr Trump's tenure in the White House, notes Professor Joseph Nye, also a former dean of the Kennedy school.
"It would be a mistake to think that the cause of this shift is Trump. There was a fire that was smouldering... Trump is like the man who comes along and pours gasoline on the fire," he tells The Sunday Times.
A report published in February by a special China task force set up by the New York-based Asia Society, chaired by seasoned China policy hands Orville Schell and Susan Shirk, sums up this new mood in Washington starkly: "The US and China are on a collision course. The foundations of goodwill that took decades to build are rapidly breaking down.
"Many Americans are starting to see China as a rising power seeking unfairly to undercut America's economic prosperity, threaten its security, and challenge its values, while many Chinese see the US as a declining power seeking to prolong its dominance by unfairly containing China's rise."
The report accuses China of "actions that defy norms of fair economic competition, abrogate international law, and violate fundamental principles of reciprocity".
It concludes: "The Trump administration is justified in pushing back harder against China's actions, but pushback alone isn't a strategy."
In addition to becoming a responsible stakeholder in the global order, many Western liberals had also harboured hopes that economic progress would push China to become "more like us", with political reforms following inevitably.
Their disappointment that this has not happened is well captured in a much-talked-about book by Mr Robert Kagan, senior fellow at the Brookings Institution, The Jungle Grows Back: America And Our Imperilled World.
Mr Kagan says: "Where once many hoped that all nations of the world would converge on a liberal democratic capitalist development, we now see authoritarianism surviving, if not thriving.
"Where once we believed that economic progress must eventually require political liberalisation, we now see autocracies successfully practising a state capitalism compatible with repressive government."
The idea that economic growth would push China towards becoming more liberal and democratic was not one that was universally subscribed to in Asia. Singapore's Mr Lee, for one, often countered that view by asserting that rather than aspiring to become a liberal democracy, the Chinese people were seeking progress and development, in pursuit not only of better lives materially, but also a sense of pride and national dignity.
A belated recognition of this is now taking hold in Western policy circles, causing some discomfort, even dismay.
Self-fulfilling prophesy?
Not everyone, however, shares Prof Allison's conclusions about what history tells us about the future of great power rivalry.
Prof Nye, for one, holds a rather different view about the future of Sino-US relations from his Harvard colleague. China, argues Prof Nye, is not about to overtake the United States any time soon, economically or technologically.
The US has many advantages, from the soft power it holds around the world, which boosts the strength of its partnerships and alliances, to its ability to draw talent from around the world to the US.
He recalls once asking Mr Lee, with whom he served on the board of French oil giant Total, if he believed that China might one day displace the US.
Mr Lee's answer, according to Prof Nye, was that while China would give the US a run for its money, given its huge market and population, it would run up against America's ability to tap the talents of billions of people around the world, combining their ideas in innovative ways.
Prof Nye concludes: "The problem for the US and China is not just the rise of power of China, it's rather the fear in the US... So, what worries me about books with titles like Destined For War, is that they contribute to the problem they are diagnosing."
Another veteran China watcher, National University of Singapore professor Wang Gungwu, also advises caution about the historical parallels drawn by Prof Allison.
"Athens and Sparta were close neighbours and basically the same people; like brothers fighting, or the bitterness of civil wars.
"Ditto with Germans and the English ruling class and most of Allison's examples.
"I am not sure US and China relations have much in common. What is real is that China is trying to recover from a disastrous 150 years and thinks that the US prefers to see the country weak and divided.
"It sees itself as trying hard not to be provoked, while the US is pushing to provide an excuse to put China down. To the Chinese leaders, if the US is setting the 'trap', they will try to avoid falling into it.
"Given the nature of war today, the wise will try their utmost to manage the really dangerous risks."
Frenemies?
Amid the escalating Sino-US trade spat, with hawks in Washington ratcheting up the rhetoric about a "decoupling" of the US and China economies, and even talk of a "clash of civilisations" with rival political systems coming head-to-head, it is little wonder that musings about the possibility of war might strike some as needlessly alarmist.
But Prof Allison argues that it is precisely the souring of bilateral ties caused by the ongoing trade dispute that risks creating the politics, perceptions and psychology that make a clash harder to avoid.
History, he says, simply shows the past patterns. The key question is whether today's political leaders will have the wisdom to learn the lessons that might be drawn from the past.
That, he says, is his next big project. He is calling for thinkers and players around the world to exercise "strategic imagination" and come up with ways to manage the "systemic risk arising from a structural reality".
Doing so calls for some deep thinking about where tensions might flare up, such as by North Korea or in the Taiwan Straits, or even the South China Sea, and working out pragmatic protocols that might enable such flashpoints to be avoided, managed and, if need be, defused.
He adds that he has been studying nine possible paths to escape the dreaded Thucydides trap. Each of these has its own lessons to be drawn, but none is exactly what he is looking for, so his search for ideas continues (see sidebar).
So far, the most promising of these, he says, is the Chanyuan treaty in 1005 between Song dynasty rulers and a proto-Mongolian tribe called the Liao, when both sides agreed to be "rivalry partners". Or, in today's parlance, "frenemies".
"They agreed to be rivals and also be partners. That sounds complex, even contradictory. But, in life, we have many such complex relationships," he says, adding that the pact gave rise to 120 years of peace.
Next, he points to the Cuban missile crisis in 1962, when President John F. Kennedy went head-to-head with his Soviet counterpart Nikita Khrushchev, and came close to plunging the world into a nuclear exchange.
Notes Prof Allison: "Kennedy came away from that experience a changed man. And he said, 'We can't do that again'. He forged a new vision. To settle for a world that was safe for diversity".
Given the tectonic shifts taking place in the world today with the rise of China, perhaps it is time to think of ways to "make the world safe for diversity", he adds, proffering the idea of "rivalry partnership" again.
This idea, he notes, is well understood in the business world. Apple and Samsung, for example, compete aggressively in the market, yet Samsung is also a components supplier to its American rival.
Prof Nye takes a similar view, and refers to what he terms "cooperative rivalry". China and the US need to realise that, despite their differences, there are many issues on which they can - and, indeed, need to - work together. These range from managing the global economy to tackling climate change.
He says: "The US and China are not existential threats to one another. We are not a threat to their existence and they are not to our existence. As long as it's on that level, then we can manage a cooperative rivalry."
Pointing to the challenge of global warming, he adds: "The present US President is not interested in climate change. But the next one, whether in 2020 or 2024, is going to have to be."
Such an approach of working together on common challenges, even while competing in other areas, would diffuse some of the inherent tensions between great powers more used to taking a zero-sum view of the world.
And doing so might help avoid the dangerous dynamic whereby underlying suspicions give rise to actions, and counter-reactions, which - as history shows - can lead to unexpected and unintended conflicts.