Here's some Rothbard:

The crucial economic question, and one of the most important social questions, is the allocation of resources: where should the various and numerous productive factors: land, labor, or capital, be allocated, and how much of each type to each use? This is the “economic problem,” and all social questions must deal with it.

The important question of American science and technology is also a problem of the allocation of resources. Thus: our expanding technology and productivity require a great many scientists, researchers, engineers, etc. It also requires many different types of resources to be invested in research and development. But our economy also requires many, many other goods and services, and many other types of investment, all of which are essential to its smooth functioning. It requires, for example, transportation to move goods, production lines to manufacture them, telephone operators and repairman to staff our giant communications network. It even requires paper manufacturers and paper distributors—for how can a modern economy—including a scientific research staff operate without paper? These are just some of the infinite number of goods and services that go to make up a functioning economy.

This fact of reality, then, must be faced: if there are to be more scientists, or more scientific research, then there must be less people and less resources available for producing all the other goods and services of the economy. The crucial question, then, is: how much? How many people and how much capital are to be funneled into each of the various occupations, including science and technology?

The problem is not that the government can’t physically fund scientists or scientific research. It obviously can, and does. But without the pricing system inherent in a market, how could a politician or bureaucrat possibly know how much money, and which resources, should fund not just science overall, but which specific areas of science and technology, and in what amount? How should research be pursued? What areas truly are the most vital to research?

The politician or bureaucrat – even in the rare case that he may be well-intentioned – has no method whatsoever to determine such things other than mere guesses. As “educated” as those guesses may be, they are still guesses. A free market does not need to make guesses. The point where the availability of certain resources intersects with the needs and wants of individuals in the market place creates the price of a certain good or service. Only these prices can guide humans to make the proper decisions regarding production of any good or service, whether it’s chicken wings, bath soap, or scientific research.

One of the great, if often unsung, merits of the free enterprise economy is that it alone can insure a smooth, rational distribution and allocation of productive resources. Through the free price systems, consumers signal laborers, capitalists, and businessmen on which occupations are most urgently needed, and the intricate, automatic workings of the price system convey these messages to everyone, thereby creating an efficient, smoothly working economy. There is one and only one alternative to voluntary direction under a free price system: and that is government dictation. And this dictation is not only bad because it violates the tradition of individual freedom and free enterprise on which American greatness is built; it is also bad because it is inevitably inefficient and self-destructive. For while government intervention can and does hamper the economic system in its job of satisfying consumer demand, it cannot force the economy to follow its own demands efficiently. For piecemeal government intervention can only disrupt an economy and defeat its own ends; while overall central planning, by destroying the price system, robs itself of the possibility of rational economic calculation. Lacking a free price system, it cannot ever satisfy the desires of either consumers or its own planners, for it will not be able to allocate the infinite number and types of labor and capital resources with any degree of efficiency.

the vast majority of research spending is done through private hand.

Here's a lecture dealing with the subject head on for 90 minutes: https://www.youtube.com/watch?v=C_PVI6V6o-4

What did the leftist anarchist say by the way? I've tried to lure out specific details or theories about the underlying mechanisms of their proposed society, but I find little but emotional appeals and half-thought-out theories.

I think most AnCap favor the right to own guns (correct me if I'm wrong). So, it is possible to have people using this power to control others. How could a society like this exist without a lot of "civil wars"?

Violence is expensive and inefficient. It would be a society filled with people, and people don't like violence, especially when they have to pay for that violence, and perform it themselves. You would also have protection agencies enforcing law, which would undoubtedly make it illegal to coerce people with firearms.