Thanks ikegami, Thanks moritz.

I have modified the code according to your comments. It is now as follows:

pl@nereida:~/Lperltesting$ cat ./comparequotedstrings.pl
#!/usr/local/lib/perl/5.10.1/bin//perl5.10.1
use v5.10;
use Benchmark qw{cmpthese};

# See http://www.regex-engineer.org/slides/img10.html
our $regexp = qr/
    ["']            # double quote
    (?:             # no memory
        [^"\\]++    # no " or escape: Don't backtrack
      | (?: \\.)++  # escaped character
    )*+             # Don't backtrack
    ["']            # end double quote
/x;

our $bregexp = qr/
  ["']            # double quote
  (?:             # no memory
      [^"\\]+     # no " or escape: backtrack
    | (?: \\.)+   # escaped character
  )*              # backtrack
  ["']            # end double quote
/x;

# input matches many times. Using "g" option
our $input = (q{"abc\"defg"hijk}x10000);
cmpthese( 0,
  {
    gbacktrack => sub { $input =~ /$bregexp/g },
    gpossessive  => sub { $input =~ /$regexp/g }
  }
);

#input matches a long string, no g option
$input = '"'.(q{abc\"defg}x10000).'"';
cmpthese( 0,
  {
    backtrack => sub { $input =~ /$bregexp/ },
    possessive  => sub { $input =~ /$regexp/ }
  }
);
# input does not match. Using "g" option
$input = '"'.("abcdefghijk"x10000);
cmpthese( 0,
  {
    failgbacktrack => sub { $input =~ /$bregexp/g },
    failgpossessive  => sub { $input =~ /$regexp/g }
  }
);
# Input does not match. Force the nested parenthesis
# to work.
our $quotes = q{\\"}x30;
$input = '"'.("abcdef $quotes ghijk"x1000);
cmpthese( 0,
  {
    failgbacktracknested => sub { $input =~ /$bregexp/g },
    failgpossessivenested  => sub { $input =~ /$regexp/g }
  }
);
[download]

Hope there are no more bugs.
The possesive qualifier now wins in the first case where the input fails

$input = '"'.("abcdefghijk"x10000);
[download]

Is the only case where the version with the possesive quantifiers wins.

Observe the huge difference in the final case (that also fails to match):

pl@nereida:~/Lperltesting$ ./comparequotedstrings.pl
                Rate gpossessive  gbacktrack
gpossessive 569399/s          --        -14%
gbacktrack  665929/s         17%          --
            Rate  backtrack possessive
backtrack  163/s         --        -4%
possessive 169/s         4%         --
                  Rate  failgbacktrack failgpossessive
failgbacktrack  16.6/s              --            -97%
failgpossessive  583/s           3419%              --

pl@nereida:~/Lperltesting$ ./comparequotedstrings.pl
                Rate gpossessive  gbacktrack
gpossessive 588574/s          --        -15%
gbacktrack  694595/s         18%          --
            Rate  backtrack possessive
backtrack  164/s         --        -4%
possessive 171/s         4%         --
                  Rate  failgbacktrack failgpossessive
failgbacktrack  17.3/s              --            -97%
failgpossessive  583/s           3276%              --
            (warning: too few iterations for a reliable count)
                        s/iter failgpossessivenested  failgbacktrackne
+sted
failgpossessivenested     23.1                    --                 -
+100%
failgbacktracknested  4.42e-02                52042%
[download]