How large sample size should be for Ljung-Box statistic to achieve a power $\ge 0.5$ when number of lags tested is 1? How does the power fall ( assuming an AR(1) process), with increasing number of lags? The LB test is more than 40 years old, surely someone must have answered these questions by now.
A little bit of googling would have answered your question. I am posting this as an answer because the document answering the question is number 2 in the google search "power of Ljung-Box test" and this question is number 3.
The answer to your question is in graphs Exhibit 14 and Exibit 15 in page 12 of said document. The power depends on the parameter of AR(1) process, the smaller it is, the lower the power, which is perfectly natural.
You can perfectly replicate these graphs yourself, here is an example of power analysis for the AR(1) processes of another test, I think it is not that hard to adapt it for Ljung-Box test.
As for the literature the usual time series books do not mention or mention the power of Ljung-Box test briefly. This means that purely theoretical results are probably not available.