I'm not too familiar with the LSA method, so I really can't speak for it.
Both of the methods that I mentioned are conceptually different. SPEIC-EASI attempts to infer the underlying dependency graph by estimating the inverse covariance matrix.
Balance trees doesn't calculate correlation, but performs a linear transformation to map species proportions to log ratios. This allows for inference of subpopulations rather than species, while enable standard statistical approaches (linear models, mixed effects models, ...) to gain insights behind microbial community fluctuation. Here's a quick 2min
video giving the run down behind this approach in case you are interested.
Best,
Jamie