Index: src/wp-admin/includes/dashboard.php
===================================================================
--- src/wp-admin/includes/dashboard.php	(revision 34913)
+++ src/wp-admin/includes/dashboard.php	(working copy)
@@ -47,7 +47,7 @@
 	}
 
 	// QuickPress Widget
-	if ( is_blog_admin() && current_user_can( 'edit_posts' ) ) {
+	if ( is_blog_admin() && current_user_can( get_post_type_object( 'post' )->cap->create_posts ) ) {
 		$quick_draft_title = sprintf( '<span class="hide-if-no-js">%1$s</span> <span class="hide-if-js">%2$s</span>', __( 'Quick Draft' ), __( 'Drafts' ) );
 		wp_add_dashboard_widget( 'dashboard_quick_press', $quick_draft_title, 'wp_dashboard_quick_press' );
 	}
Index: src/wp-admin/post.php
===================================================================
--- src/wp-admin/post.php	(revision 34913)
+++ src/wp-admin/post.php	(working copy)
@@ -71,7 +71,7 @@
 	if ( ! wp_verify_nonce( $nonce, 'add-post' ) )
 		$error_msg = __( 'Unable to submit this form, please refresh and try again.' );
 
-	if ( ! current_user_can( 'edit_posts' ) ) {
+	if ( ! current_user_can( get_post_type_object( 'post' )->cap->create_posts ) ) {
 		exit;
 	}