How reliable is BLS / JOLTS data on quits?
Here is a short lists of study’s that looks at the on the reliance of JOLTS data as well as make comparisons to other similar data sets.

Analysis of JOLTS Research Estimates by Size of Firm October 2012- 
 Katherine Bauer Klemmer
https://www.bls.gov/osmr/pdf/st120060.pdf

ADJUSTED ESTIMATES OF WORKER FLOWS AND JOB OPENINGS IN JOLTS-  Steven Davis,  R. Jason Faberman, John Haltiwanger, Ian Rucker (2018).
http://faculty.chicagobooth.edu/steven.davis/pdf/MS_ch5_StevenDavis-et-al_p315-370.pdf

Help-Wanted Advertising and Job Vacancies- Rob Valletta – Federal Reserve San Francisco
https://www.frbsf.org/economic-research/publications/economic-letter/2005/january/help-wanted-advertising-and-job-vacancies/ 

UNDERSTANDING ONLINE JOB ADS DATA  - Anthony Carnevale, Tamara Jayasundera, Dmitri Repnikov 
https://cew.georgetown.edu/wp-content/uploads/2014/11/OCLM.Tech_.Web_.pdf

Reliability of JOLTS Estimates
https://www.bls.gov/jlt/jltreliability.htm