Socialism

Cuba, China, and Vietnam are all great examples to study. DPRK is doing well given that it's completely blockaded by the global empire. Meanwhile, US has poured untold billions into propping SK up. Your argument completely ignores the fact that every communist country has to content with the aggression from the most powerful global empire to ever exist. It's not like these countries are allowed to develop peacefully. Soviet Union provided a counterbalance and protected these countries from aggression by the US. Once it dissolved US redoubled its efforts to destroy these countries. If anything, the very fact that they are still around and thriving shows how resilient communism.

Meanwhile, adopting capitalist policies doesn't make a country capitalist any more than adopting some social services makes a country communist. What makes a country communist or capitalist is determined by which class holds power.

The necessarily government represents the interests of the class that holds power. In a capitalist society that happens to be the capital owning class. This is why you see laws and regulations that inhibit monopolization erode over time. For example, US has very strong worker protections and laws ensuring fair competition after the New Deal was passed. All of that was then dismantled, and here's what a recent study analyzing many decades of US policy has to say about the state of things today:

What do our findings say about democracy in America? They certainly constitute troubling news for advocates of “populistic” democracy, who want governments to respond primarily or exclusively to the policy preferences of their citizens. In the United States, our findings indicate, the majority does not rule—at least not in the causal sense of actually determining policy outcomes. When a majority of citizens disagrees with economic elites or with organized interests, they generally lose. Moreover, because of the strong status quo bias built into the U.S. political system, even when fairly large majorities of Americans favor policy change, they generally do not get it.

You didn’t answer my question though, you just gave a list of countries (some of which, like China, I question are socialist at all). Are there any countries where “working class holds power, and the core economy of the country is either publicly or cooperatively owned” that have out-innovated countries that use capitalism?

I did answer your question though. All these countries, including China, are socialist because it's the working class that holds power. We can look at the tangible outcomes in China such as poverty reduction programs that simply aren't happening in countries where capitalists are in charge. So, we don't have to take their word for it, we can just look at the outcomes. Let's look at China in a bit more detail though.

First, it's worth noting that 87.6% of young Chinese identify with Marxism, and the party has 95 million members. I think it's reasonable to assume that people of a country where vast majority of young people identify as Marxist understand what sort of a political system their country has.

All the essential industry is state owned, and capitalism is only allowed to exist within special economic zones. However, even Marx argued that capitalism is likely a necessary stage for developing productive resources needed for socialism and communism to be possible. Arguing that capitalism being allowed in China makes it capitalist would be akin to arguing that having social services such as public healthcare makes Canada communist.

One simple test to consider is that China doesn't suffer from regular crashes seen under capitalism. An inherent contradiction within capitalism is that the capitalists always want to cut pay for their employees to minimize the costs, while they also require consumers with enough spending power to consume the commodities they produce. This is why capitalism results in regular economic crashes when wages fall below the point where consumption can keep up with the rate of commodity production. At that point you end up with overproduction and a crash. If China was capitalist then it should be experiencing these kinds of crashes regularly just like actual capitalist nations are in the Western world.

And a related point is that quality of life in China continues to steadily improve and the government is actively working on doing things like eliminating poverty, creating public infrastructure, providing healthcare, housing, food, and education for all citizens. Chinese government practically eliminated poverty, and in fact China is the only place in a world where any meaningful poverty reduction is happening. If we take China out of the equation poverty actually increased in real terms:

If we take just one country, China, out of the global poverty equation, then even under the $1.90 poverty standard we find that the extreme poverty headcount is the exact same as it was in 1981.

The $1.90/day (2011 PPP) line is not an adequate or in any way satisfactory level of consumption; it is explicitly an extreme measure. Some analysts suggest that around $7.40/day is the minimum necessary to achieve good nutrition and normal life expectancy, while others propose we use the US poverty line, which is $15.

Real wage (i.e. the wage adjusted for the prices you pay) has gone up 4x in the past 25 years, more than any other country. This is staggering considering it's the most populous country on the planet. Social mobility in China is actually higher than it is in US. The opposite is currently happening in capitalist countries. In particular, it's instructive to look at the differences in development between China and India with both having started roughly in the same place and having comparable population.

Another indicator is that China used more concrete in 3 years than US in all of 20th century, they built 27,000km of high speed rail in a decade. This is another thing we don't see happening under capitalism because capitalists don't see significant profit from infrastructure investments. This is the main reason US infrastructure is currently crumbling.

Finally, 90% of families in the country own their home giving China one of the highest home ownership rates in the world. What’s more is that 80% of these homes are owned outright, without mortgages or any other leans. This sort of home ownership is not seen in capitalist countries where housing has become a commodity.

What is the primary force that spurs innovation in a socialist country and how does it compare to the force that spurs innovation in a capitalist country?

People wanting to do interesting things because they're curious and intelligent, The same force that has driven innovation throughout history. Are you seriously not to be able to imagine having any sort of hobby or interest that's not profit driven. Why do you think millions of people around the world create innovative open source projects without any profit motive. You're literally using an innovative federated platform built by volunteers to ask this inane question.