How is Q-carbon made? 
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*How is Q- carbon made ?
https://en.wikipedia.org/wiki/Q-carbon

*Why a nanosecond laser is needed?

*How is it cooled ?
 A: The original paper was published in J. Appl. Phys. 118, 215303 (2015) and was accompanied by a NCSU news report. 
I do not have access to J.Appl.Phys., but the news report provides a reasonable answer to the first 2 questions: 

  
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*How is Q-carbon made?
  

"Researchers start with a substrate, such as such as sapphire, glass or a plastic polymer. The substrate is then coated with amorphous carbon – elemental carbon that, unlike graphite or diamond, does not have a regular, well-defined crystalline structure. The carbon is then hit with a single laser pulse lasting approximately 200 nanoseconds. During this pulse, the temperature of the carbon is raised to 4,000 Kelvin (or around 3,727 degrees Celsius) and then rapidly cooled. This operation takes place at one atmosphere – the same pressure as the surrounding air.
The end result is a film of Q-carbon, and researchers can control the process to make films between 20 nanometers and 500 nanometers thick."


  
*Why a nanosecond laser is need?
  

For fast heating, see above. From what transpires in the same source, the laser pulse brings the amorphous carbon film to liquid state, then rapid cooling, or quenching, produces the new carbon phase. A related piece of info: 
"By using different substrates and changing the duration of the laser pulse, the researchers can also control how quickly the carbon cools. By changing the rate of cooling, they are able to create diamond structures within the Q-carbon."


  
*How is it cooled? 
  

Not sure, but since the process takes place at room temperature and pressure, air/gas flow seems likely. 
