text: We next test the Twigstats time-restricted genealogy approach 
  on a range of empirical examples. First, we boost pairwise outgroup 
  *f*3-statistics44 to quantify fine-scale population structure; we demon- 
  strate this improvement using a previously proposed simulation39 
  (Extended Data Fig. 5a). When applied to published genomes from 
  Neolithic Europe (Methods and Supplementary Table 1), we can repli- 
  cate the previously suggested fine-scale structure between individuals 
  buried in megalithic structures in Ireland compared with others45, a 
  relationship that is not apparent from SNP data alone (Extended Data 
  Fig. 5b). For the well-studied example of three major ancestries contrib- 
  uting to prehistoric Europe, that is, Mesolithic hunter-gatherers, early 
  farmers and steppe populations13–16, we obtain unbiased estimates and 
  an approximately 20% improvement in standard errors in an already 
  well-powered qpAdm model46 (Extended Data Fig. 5c). 
  
  Finally, we demonstrate that Twigstats can be used to resolve com- 
  peting models of punctual admixture and long-standing gene flow, or 
  constrain the time of admixture. For instance, it has previously been 
  suggested that long-standing deep structure and gene flow between 
  Neanderthals and early modern humans in Africa may produce genetic 
  patterns that resemble a punctual admixture event some 60,000 years 
  ago47–49, casting doubt on the model of Neanderthal admixture into 
  ancestors of Eurasians49–51. However, whereas such long-standing deep 
  substructure would confound SNP-based*f*-statistics to produce pat- 
  terns similar to Neanderthal admixture, we demonstrate, in simula- 
  tions, that Twigstats can clearly distinguish this history from recent 
  admixture (Extended Data Fig. 5d). Application of Twigstats on empiri- 
  cal whole genomes produces results inconsistent with deep substruc- 
  ture alone, but consistent with punctual admixture. 