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How can I for example convert {"a","b"} to "{\"a\",\"b\"}", so keeping the quotations marks.? I tried ToString[{"a","b"}], but this returns "{a,b}" in Mathematica 11.2. Thanks!

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ToString[{"a", "b"}, InputForm]
(*    "{\"a\", \"b\"}"    *)

(thanks @kglr for golfing!)

If you need to get rid of the space between the strings in the string (as per your specification), then use a StringDelete[" "] operator:

ToString[{"a", "b"}, InputForm] // StringDelete[" "]
(*    "{\"a\",\"b\"}"    *)
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    $\begingroup$ StringDelete[" "] wold also delete any " " inside the quotation marks, wouldn't it? $\endgroup$ – rhermans Jul 18 at 9:57
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    $\begingroup$ @rhermans yes you're right, may be a bad idea. $\endgroup$ – Roman Jul 18 at 9:58
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    $\begingroup$ ToString[{"a", "b"}, InputForm] also works. (+1) $\endgroup$ – kglr Jul 18 at 10:14
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ToString[{"a", "b"} /. x_String :> ("\"" <> x <> "\"")] seems to work.

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StringTemplate["{\"``\",\"``\"}"] @@ {"a", "b"}

"{\"a\",\"b\"}"

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There is a short :) function to do this:

StringJoin @ Flatten @ ReplaceAll[RowBox -> List] @  
First @ First @ MathLink`CallFrontEnd @ UndocumentedTestFEParserPacket @ 
ToString @ InputForm @
{"a", "b"}

The idea is to create input form string with quoted characters, then to parse it as an input line (which will remove spaces) then take remaining strings and join them :)

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