Democratic communism is different from republican capitalism, so it is definitely the case that the kinds of innovation will differ between communism and capitalism. Obviously, any potential entrepreneur who requires the possibility of staggering wealth to create his innovation will not have that motivation under communism. More importantly, the general conditions of winner take all and innovate or die under capitalism will be greatly relaxed. Generally speaking, neither people nor firms will work as hard at something if it is no longer a matter of life and death. I expect that innovation under communism will occur at a much less frenetic pace as under 19th and 20th century capitalism.
*In the communist phase, the material conditions of society are developed to such a degree that relative wealth becomes as meaningless for ordinary people as relative access to air.
**I mean "apologist" in the neutral sense, as someone who argues for a position.
***The bias here towards technology entrepreneurs is partly due to my own history as a computer programmer.
But slower is not stopped. Even a cursory study of history shows that many people have contributed substantial innovations without the prospect of enormous wealth. Academic scientists and scholars have created numerous innovations, from relativity to educational methods to critical race theory, without the realistic possibility of enormous wealth. People sometimes act like the iPod sprang Athena-like from the head of Steve Jobs, but in reality Apple's innovative products are the work of many thousands of engineers and technicians — not to mention all the people supporting their efforts: even the most brilliant engineer is going to have a difficult time innovating if the bathrooms are filthy. I use many free software packages, not just prestige products such as Firefox, but a host of innovative and useful applications created by people who just wanted to see something cool in the world. Capitalists are just as prone to ethnocentrism as anyone else: they themselves are motivated by solely by wealth, so they tend to think that everyone is motivated solely by wealth. But of course people have a lot of different motives, and the urge to create is a basic human desire, and people will innovate even if there were no social incentives at all.
We also must look critically at the kinds of innovation that capitalism supports: capitalism supports only those innovations that have the potential of making some capitalists relatively wealthier, compared to each other and compared to the professional/managerial and working classes. This structure is problematic in a variety of ways. First, innovation is concentrated on private goods, and innovation in public goods either languishes or is actively blocked. Hence we have Viagra and Rogaine, but research on new antibiotics is practically nil. Innovations in health care financing are actively blocked, as demonstrated by the absolutely unnecessary tsuris over the PPACA, when the obvious answer — not even innovative anymore — is the expansion of Medicare. Second, innovations in what are more or less "naturally" public — non-exclusive, non-rival — goods require that we attempt to make them into private — exclusive, rival — goods. The most obvious example is copyright law. In the electronic age, movies and computer programs are "naturally" non-exclusive and non-rival; copyright law tries, by brute force, to make these goods exclusive and rival. Finally, some "innovations" focused on capitalist wealth, such as recent "innovations" in the financial system, have proven to be unmitigated social disasters, making everyone but a few vastly worse off, and nearly wrecking the global economy. There are many capitalist innovations that few would miss if their motivations were removed.
In contrast, communism stresses innovation in public rather than private goods. Allocation of capital under communism is directly democratic. Under capitalism, the entrepreneur must convince people who are already rich that her innovation will make them richer; under communism, the entrepreneur must convince citizens that her innovation will make them better off. Critics of communism thus must either "bite the bullet" and say that democracy itself is fundamentally evil or ineffective; they must say that what will garner the approval of a democratic majority is not best, or that citizens do not and cannot know what will make them better off. Assuming democracy is not an evil, then the objection of capitalist apologists is easily met: entrepreneurs will innovate along lines that will gain the approval of the majority, and the majority will innovations they do not want.
Finally, I want to reiterate a point that I've made many times earlier: communism cannot and will not actually happen until it is clear that capitalism itself has become a fetter on the means of production. Revolutionary ideology does not and cannot in good conscience advocate replace a system that is working, however imperfectly, with an untested system whose foundation is purely theoretical; not only is such advocacy morally wrong, its goal is practically impossible. As long as apologists and reformers can keep the system working, however imperfectly, they will not fall.* However, when the "innovations" of capitalism nearly always do not have the effect of increasing the public good but only maintaining the power and privilege of the capitalist ruling class, communists have the opportunity of introducing a newer, better paradigm of innovation.
*There is more scope for changing a merely imperfect system in the alternative mode of revolutionary change, "allopatric" revolution, similar to the concept of allopatric speciation in biological evolution.
Let me close with a quotation from J. S. Mill, from Principles of Political Economy:
I confess I am not charmed with the ideal of life held out by those who think that the normal state of human beings is that of struggling to get on; that the trampling, crushing, elbowing, and treading on each other's heels, which form the existing type of social life, are the most desirable lot of human kind, or anything but the disagreeable symptoms of one of the phases of industrial progress. It may be a necessary stage in the progress of civilization, and those European nations which have hitherto been so fortunate as to be preserved from it, may have it yet to undergo. It is an incident of growth, not a mark of decline, for it is not necessarily destructive of the higher aspirations and the heroic virtues; as America, in her great civil war, has proved to the world, both by her conduct as a people and by numerous splendid individual examples, and as England, it is to be hoped, would also prove, on an equally trying and exciting occasion. But it is not a kind of social perfection which philanthropists to come will feel any very eager desire to assist in realizing. Most fitting, indeed, is it, that while riches are power, and to grow as rich as possible the universal object of ambition, the path to its attainment should be open to all, without favour or partiality. But the best state for human nature is that in which, while no one is poor, no one desires to be richer, nor has any reason to fear being thrust back by the efforts of others to push themselves forward. (bk. IV, ch. VI sec. 2)