I expect that the corresponding nested if..else construct would be even more baffling. The weakness here is in the negated and chopped-up logic, not the trinary op. I'll make a first cut at rewriting it:

my $send_now_button =
    $auth{'SEND'} && $status==1 && $approved && !$send_now ?
    qq( ...blah blah ...) : '';
[download]

Not nested, you can use it now ;-)

I think I had an easier time decyphering the (well-formatted!) nested trinary than I would have a similarly built if statement.

Update: Just for fun, here's thing I'm slanging, formatted the best I know how,

my $send_now_button;
if ( !$auth{'SEND'} ) {
    $send_now_button = '';
}
else {
    if ( $status != 1 ) {
        $send_now_button = '';
    }
    else {
        if ( $approved ) {
            if ( $send_now ) {
                $send_now_button = '';
            }
            else {
                $send_now_button = qq( ...blah blah ...);
            }
        }
        else {
            if ( $send_now ) {
                $send_now_button = '';
            }
            else {
                $send_now_button = '';
            }
        }
    }
}
[download]

Noisy, ain't it? That maybe does even better than the trinary version at pointing out how ludicrous the logic is, but IMO trinary makes it easier to see how to fix it. I didn't even bring elsif to the table :-))