在Ant Design框架中，可以通过antd的Checkbox组件来实现单选和多选的功能。其中，多选时需要将多个checkbox放在Checkbox.Group中。

要保存状态，可以使用React中的useState hook来记录选中的选项。当用户改变选择时，将会更新状态，并将其保存在本地存储中。

下面是一个示例代码：

import React, { useState } from 'react';
import { Checkbox } from 'antd';

const CheckboxGroup = Checkbox.Group;

function CheckboxDemo() {
  const [checkedList, setCheckedList] = useState([]);
  const [indeterminate, setIndeterminate] = useState(true);
  const [checkAll, setCheckAll] = useState(false);

  const plainOptions = ['Apple', 'Pear', 'Orange'];
  const options = [
    { label: 'Apple', value: 'Apple' },
    { label: 'Pear', value: 'Pear' },
    { label: 'Orange', value: 'Orange' },
  ];

  const onChange = (list) => {
    setCheckedList(list);
    setIndeterminate(!!list.length && list.length < options.length);
    setCheckAll(list.length === options.length);
    localStorage.setItem('checkboxList', JSON.stringify(list));
  };

  const onCheckAllChange = (e) => {
    setCheckedList(e.target.checked ? plainOptions : []);
    setIndeterminate(false);
    setCheckAll(e.target.checked);
    localStorage.setItem('checkboxList', JSON.stringify(e.target.checked ? plainOptions : []));
  };

  useEffect(() => {
    const storedList = JSON.parse(localStorage.getItem('checkboxList'));
    if (storedList) {
      setCheckedList(storedList);
      setIndeterminate(!!storedList.length && storedList.length < options.length);
      setCheckAll(storedList.length === options.length);
    }
  }, []);

  return (
    

      
        Check all
      
      
    
  );
}

export default CheckboxDemo;